Prospect Theory: For Risk and Ambiguity

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Prospect Theory

Prospect Theory: For Risk and Ambiguity provides the first comprehensive and accessible textbook treatment of the way decisions are made both when we have the statistical probabilities associated with uncertain future events (risk) and when we lack them (ambiguity). The book presents models, primarily prospect theory, that are both tractable and psychologically realistic. A method of presentation is chosen that makes the empirical meaning of each theoretical model completely transparent. Prospect theory has many applications in a wide variety of disciplines. The material in the book has been carefully organized so as to allow readers to select pathways relevant to their own interests. With numerous exercises and worked examples, the book is ideally suited to the needs of students taking courses in decision theory in economics, mathematics, finance, psychology, management science, health, computer science, Bayesian statistics, and engineering. A companion website with exercises, additional assignments, and solutions is available at www.cambridge.org/wakker. peter p. wakker is Professor at the Econometric Institute, Erasmus University. He has worked in departments of mathematics, statistics, psychology, business, medicine, and economics in the Netherlands and in the USA, and has published in leading journals in all of these fields. He serves on the editorial boards of several major journals, including Journal of Behavioral Decision Making, Journal of Mathematical Psychology, Journal of Risk and Uncertainty, Management Science, Medical Decision Making, and Theory and Decision.

Prospect Theory For Risk and Ambiguity

Peter P. Wakker

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521765015 © Peter P. Wakker This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-77657-1

eBook (NetLibrary)

ISBN-13

978-0-521-76501-5

Hardback

ISBN-13

978-0-521-74868-1

Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Introduction Prospect theory Behavioral foundations Homeomorphic versus paramorphic modeling Intended audience Attractive feature of decision theory Structure Preview Our five-step presentation of decision models Part I

page xiii 1 2 2 3 3 4 4 5 7

Expected utility 1

The general model of decision under uncertainty and no-arbitrage (expected utility with known utilities and unknown probabilities) 1.1 Basic definitions 1.2 Basic properties of preferences 1.3 Expected value 1.4 Data fitting for expected value 1.5 The bookmaking argument of de Finetti, or the no-arbitrage condition from finance 1.6 A behavioral foundation of subjective probabilities and expected value using no-arbitrage and no-book 1.7 Discussion of Structural Assumption 1.2.1 (Decision under uncertainty) 1.8 The general nature and usefulness of behavioral foundations 1.9 Appendix: Natural, technical, and intuitive preference conditions, and the problematic completeness condition 1.10 Appendix: The relation of Theorem 1.6.1 to the literature 1.11 Appendix: Proof of Theorem 1.6.1 and Observation 1.6.10

11 11 15 17 19 22 27 32 34 37 39 41 v

Contents

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2

Expected utility with known probabilities – “risk” – and unknown utilities 2.1 Decision under risk as a special case of decision under uncertainty 2.2 Decision under risk: basic concepts 2.3 Decision under risk as a special case of decision under uncertainty; continued 2.4 Choices under risk and decision trees 2.5 Expected utility and utility measurement 2.6 Consistency of measurement and a behavioral foundation of expected utility 2.7 Independence and other preference conditions for expected utility 2.8 Basic choice inconsistencies 2.9 Appendix: Proof of Theorem 2.6.3 and Observation 2.6.30

63 65 67

3

Applications of expected utility for risk 3.1 An application from the health domain: decision tree analysis 3.2 Risk aversion 3.3 Applications of risk aversion 3.4 Indexes of risk aversion 3.5 Parametric families of utility 3.6 Data fitting for expected utility under risk 3.7 Multiattribute utility 3.8 Taxation and welfare theory with cardinal utility 3.9 Proofs for Chapter 3

69 69 71 74 77 78 83 85 91 92

4

Expected utility with unknown probabilities and unknown utilities 4.1 Fifteen preference questions 4.2 (Subjective) expected utility 4.3 Measuring utility and testing EU for }4.1 4.4 A decision process 4.5 General utility measurement and the tradeoff notation 4.6 A behavioral foundation of expected utility 4.7 Further discussion of Theorem 4.6.4 (Uncertainty-EU) 4.8 Further implications of expected utility 4.8.1 Savage’s sure-thing principle 4.8.2 Utility analysis based on t-indifferences 4.8.3 Debreu’s additive decomposability and state-dependent utility 4.9 A hybrid case: subjective probabilities and utilities when also objective probabilities are given 4.9.1 A general (“single-stage”) approach

44 44 47 48 51 53 57

94 94 100 103 107 108 110 112 114 114 115 117 118 119

4.10 4.11 4.12 4.13 4.14 4.15 Part II 5

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Contents

vii

4.9.2 A behavioral foundation for the single-stage approach 4.9.3 The multi-stage approach of Anscombe & Aumann (1963) 4.9.4 Comparing the Anscombe & Aumann model with the single-stage approach Data fitting for expected utility under uncertainty A statistical analysis of the experiment of }4.1, revealing violations of expected utility The Allais paradox: a well-known violation of expected utility Appendix: Experimental economics’ real-incentive principle and the tradeoff method Appendix: Tradeoff consistency as a generalization of de Finetti’s bookmaking Appendix: Proofs for Chapter 4

120 122 125 127 131 133 135 138 139

Nonexpected utility for risk Heuristic arguments for probabilistic sensitivity and rank dependence 5.1 Probabilistic sensitivity versus outcome sensitivity for single-nonzero-outcome-prospects 5.2 Probabilistic sensitivity for multi-outcome prospects, and the old way of transforming probabilities 5.3 A violation of stochastic dominance 5.4 Rank-dependent utility: discovering the formula using psychological intuition 5.5 Rank-dependent utility: discovering the formula using mathematical intuition 5.6 Calculating rank-dependent utility 5.7 Conclusion Probabilistic sensitivity and rank dependence analyzed 6.1 Ranks, ranked probabilities, and rank-dependent utility defined 6.2 Ranks, ranked probabilities, and rank-dependent utility discussed 6.3 Where rank-dependent utility deviates from expected utility: optimism and pessimism 6.4 Further deviations of rank-dependent utility from expected utility 6.4.1 The certainty effect 6.4.2 Further discussion of rank dependence 6.5 What rank-dependent utility shares with expected utility 6.5.1 Measuring utility under rank-dependent utility 6.5.2 A behavioral foundation of rank-dependent utility 6.5.3 An elaborated example 6.5.4 Abdellaoui’s method of measuring probability weighting 6.5.5 Further implications of rank-dependent utility

145 145 149 153 155 161 165 168 169 169 172 172 176 176 178 181 181 184 185 188 188

Contents

viii

6.6 Appendix: Yet further deviations of rank-dependent utility from expected utility 6.7 Appendix: Ranking identical outcomes and collapsing outcomes 6.8 Appendix: An interpretation of rank-dependent utility using the derivative of w 6.9 Appendix: RDU for continuous distributions and figures for negative outcomes 7

8

Applications and extensions of rank dependence 7.1 Likelihood insensitivity and pessimism as two components of probabilistic risk attitudes 7.2 Parametric forms of weighting functions 7.3 Data fitting for rank-dependent utility under risk 7.4 Direct ways to test convexity, concavity, and likelihood insensitivity using violations of the sure-thing principle 7.5 An alternative direct way to directly investigate properties of nonlinear decision weights 7.6 Bad-news probabilities or loss-ranks 7.7 A formal definition of likelihood insensitivity (inverse-S) 7.8 The choice of insensitivity region 7.9 Discussion of likelihood insensitivity 7.10 Indexes of pessimism and likelihood insensitivity 7.11 Binary rank-dependent utility 7.12 Appendix: Alternative definition of likelihood insensitivity: first concave and then convex 7.13 Appendix: Proofs for Chapter 7 Where prospect theory deviates from rank-dependent utility and expected utility: reference dependence versus asset integration 8.1 A choice paradox 8.2 A discussion and the real culprit of the paradox 8.3 Deviations from a fixed reference point (“initial wealth”) as nothing but an alternative way of modeling final wealth, and of incorporating some empirical improvements 8.4 Loss aversion defined 8.5 Deviations from a variable reference point as a major breakaway from final-wealth models 8.6 Rabin’s paradox 8.7 Future directions for theories of reference dependence 8.8 Appendix: Empirical meaningfulness problems of loss aversion 8.9 Appendix: A formal model of initial wealth and reference points

190 194 197 199 203 203 206 211 214 217 219 222 226 227 229 230 231 233

234 235 236

237 238 240 242 245 247 249

9

Part III 10

Contents

ix

Prospect theory for decision under risk 9.1 A symmetry about 0 underlying prospect theory 9.2 The definition of prospect theory 9.3 Properties of the prospect theory formula, and calculations 9.4 What prospect theory shares with rank-dependent utility and with expected utility, and one more reanalysis of the experiment of } 4.1 9.4.1 Monotonicity, stochastic dominance, and the sure-thing principle 9.4.2 Measuring utility, event weighting, and loss aversion 9.5 Empirical findings on prospect theory 9.6 Analytical problems of power utility for analyzing loss aversion 9.6.1 The first problem: loss aversion depends on the monetary unit 9.6.2 The second problem: always U(a) > U(a) for some outcome a > 0 9.6.3 Ways to avoid the problems 9.7 Appendix: Some theoretical issues concerning prospect theory 9.8 Appendix: Original prospect theory of 1979

251 251 252 254

260 260 262 264 267 267 270 271 271 273

Nonexpected utility for uncertainty Extending rank-dependent utility from risk to uncertainty 10.1 Probabilistic sophistication 10.2 Rank-dependent utility defined for uncertainty and without probabilistic sophistication 10.3 Rank-dependent utility and probabilistic sophistication once more 10.3.1 Ellsberg’s violation of probabilistic sophistication 10.3.2 Further observations 10.4 Where rank-dependent utility under uncertainty deviates from expected utility in the same way as it did under risk 10.4.1 Optimism and pessimism 10.4.2 Likelihood insensitivity (Inverse-S) 10.4.3 Direct ways to test convexity, concavity, and likelihood insensitivity using violations of the sure-thing principle 10.5 What rank-dependent utility shares with expected utility for uncertainty (in the same way as it did for risk) 10.5.1 Rank-preference conditions 10.5.2 Measuring utility (and event weighting) under RDU 10.5.3 A behavioral foundation for RDU 10.6 Binary rank-dependent utility 10.7 A hybrid case: rank-dependent utility for uncertainty when also objective probabilities are given

279 279 282 286 286 287 288 289 290 292 294 294 295 297 298 299

Contents

x

10.8 10.9 10.10 10.11 10.12 10.13 10.14 11

12

10.7.1 A general (“single-stage”) approach 10.7.2 A behavioral foundation for the single-stage approach 10.7.3 Schmeidler’s and Jaffray’s two-stage approaches Bad-news events or loss-ranks Further observations regarding rank dependence Appendix: An integral representation Appendix: Ranking states and collapsing outcomes Appendix: Comonotonicity Appendix: Proofs for Chapter 10 Appendix: Proof of Theorem 6.5.6

Ambiguity: where uncertainty extends beyond risk 11.1 The Ellsberg paradox and the home bias as within-subject between-source comparisons 11.2 Using rank dependence to analyze ambiguity 11.3 Elaborated examples for ranks, ambiguity premiums, and hedging in finance 11.4 The CORE 11.5 Multiple priors (Maxmin expected utility and a-maxmin): basic results 11.6 Approaches in the literature that we will not take 11.7 Indexes of ambiguity aversion 11.8 Indexes of ambiguity aversion and sensitivity 11.9 Appendix: Discussion of multiple priors and other models 11.10 Appendix: Violations of monotonicity for multiple priors 11.11 Appendix: Mo¨bius transforms and belief functions Prospect theory for uncertainty 12.1 Prospect theory defined 12.2 Where prospect theory deviates from rank-dependent utility and expected utility 12.3 What prospect theory shares with rank-dependent utility and expected utility under uncertainty 12.3.1 Sign-preference conditions 12.3.2 Measuring utility (and event weighting) under prospect theory 12.3.3 Measuring loss aversion under prospect theory 12.3.4 A behavioral foundation for prospect theory 12.4 A hybrid case: prospect theory for uncertainty when also objective probabilities are given 12.5 Loss aversion versus ambiguity aversion 12.6 Violations of prospect theory and rank dependence 12.7 Indexes of ambiguity attitudes under prospect theory

299 300 301 304 305 309 310 310 314 316 317 317 318 322 323 324 327 328 332 335 338 339 342 342 344 344 344 345 346 347 349 349 350 354

13

Contents

xi

12.8 Appendix: Some theoretical issues concerning prospect theory for uncertainty 12.9 Appendix: Proofs for Chapter 12

356 357

Conclusion

358

Appendices Appendix A

359

Models never hold perfectly: how to handle their deficiencies? A.1 Nonparametric measurements and parametric fittings for imperfect models: general discussion A.2 Our distance measure for parametric fitting A.3 Discussion of the distance measure Appendix B Choosing from multiple prospects and binary choice: the principles of revealed preference B.1 Examples B.2 A behavioral foundation of preference relations derived from choice behavior B.3 Assumptions underlying revealed preference B.4 The history of revealed preference Appendix C Dynamic decisions Appendix D Applications other than decision under uncertainty Appendix E Bisymmetry-based preference conditions Appendix F Nonmonotonic rank-dependent models and the Fehr–Schmidt model of welfare evaluation Appendix G Extensions of finite-dimensional results to infinite-dimensional results: a meta-theorem Appendix H Measure theory Appendix I Related textbooks and surveys Appendix J Elaborations of exercises Appendix K Skipping parts and interdependencies between sections

393 395 397 399 455

References Author index Subject index

461 492 500

359 359 361 363 366 367 371 374 378 380 384 387 391

Preface

This book is the culmination of 14 years of teaching. In the 15th year, when for the first time I did not feel like rereading or rewriting, the time had come to publish it. The book received helpful comments from Han Bleichrodt, Arie de Wild, Itzhak Gilboa, Glenn Harrison, Amit Kothiyal, Gijs van de Kuilen, Georg Weizsa¨cker, and many students during the past 14 years. Thorough comments from Rich Gonzalez and Vitalie Spinu are especially acknowledged. I am most indebted to Stefan Trautmann for the numerous improvements he suggested. This book has also benefited from many inspiring discussions with Craig Fox, with whom I share the privilege of having collaborated with Amos Tversky on uncertainty during the last years of his life.

xiii

Introduction

This book has been written and organized especially for readers who do not want to read all of its contents, but want to skip parts and select the material of their own interest. This has been achieved by an organization of exercises explained later, and by an Appendix K that describes the interdependencies between sections. Because of this organization, this book can be used by readers with different backgrounds. We will examine theories of individual decision making under uncertainty. Many of our decisions are made without complete information about all relevant aspects. This happens for instance if we want to gamble on a horse race and have to decide which horse to bet on, or if we are in a casino and have to decide how to play roulette, if at all. Then we are uncertain about which horse will win or how the roulette wheel will be spun. More serious examples include investments, insurance, the uncertain results of medical treatments, and the next move of your opponent in a conflict. In financial crises, catastrophes can result from the irrational attitudes of individuals and institutions towards risks and uncertainties. Two central theories in this book are expected utility theory and prospect theory. For all theories considered, we will present ways to empirically test their validity and their properties. In many applications we require more than just qualitative information. We may want to know exactly that spending 1 percent more on a new medicine will generate a 3 percent increase in quality of life for the patient group affected, rather than just knowing that spending more money improves the quality of life. Similarly, we may want to know that a person is willing to pay a maximum of $350 extra tax so as to avoid a 1:100 risk of losing savings to the value of $30,000 in case a bank goes bankrupt. Hence, for all the theories presented in this book, methods will be provided for obtaining precise quantitative measurements concerning those theories and their parameters. Thus precise quantitative predictions can be made. The possibility of obtaining tractable quantitative measurements was a selection criterion for the theories presented in this book. Typical for the analyses in this book is the interaction between, on the one hand, theoretical and algebraic tools, and, on the other hand:

1

2

Introduction

• prescriptive considerations as relevant for consultancies, policy decisions, and your own decision making; • descriptive considerations as relevant in psychology and other empirical disciplines. Prospect theory

Until the end of the 1970s, irrational behavior was believed to be chaotic and unsuited for modeling. The normative expected utility model was taken to be the best approximation of descriptive behavior (Arrow 1951a p. 406; Tversky & Kahneman 1981 opening sentence). Kahneman & Tversky’s (1979) prospect theory provided a major breakaway. It was the first descriptive theory that explicitly incorporated irrational behavior in an empirically realistic manner (Kahneman 2003 p. 1456), while at the same time being systematic and tractable. It was the first rational theory of irrational behavior, so to say. Tversky & Kahneman (1992) introduced an improved version of prospect theory. First, they used Quiggin’s (1982) rank dependence to correct a theoretical problem in probability weighting. Second, and more importantly, they extended the theory from risk (known probabilities) to uncertainty and ambiguity (unknown probabilities), using Schmeidler’s (1989) rank dependence. In this manner, for the first time a theory has resulted that combines empirical realism with theoretical soundness and tractability. Prospect theory comprises the happy marriage between the empirical insights of Kahneman & Tversky (1979) and the theoretical insights of Gilboa (1987) and Schmeidler (1989). At this moment of writing, 30 years after its invention, prospect theory is still the only theory that can deliver the full spectrum of what is required for decision under uncertainty, with a natural integration of risk and ambiguity. Therefore, a textbook on the theory is useful. The main purpose of this book is to make this theory accessible to a wide audience by presenting it in a manner as tractable as possible. Behavioral foundations

Behavioral foundations will play a central role in this book. For a particular decision model, a behavioral foundation gives a list of conditions, stated directly in terms of observable preferences, that hold if and only if the decision model holds. Preference foundations translate the meaning of quantitative decision models and their subjective parameters (“theoretical constructs”), such as subjective probabilities or utilities, into observables. Descriptively, they show how to verify or falsify decision models. Normatively, they provide the terms to justify or criticize models. When de Finetti (1931a), von Neumann & Morgenstern (1944), and Savage (1954) provided behavioral foundations for expected utility, this gave a big boost to the popularity of this theory in many fields. Those fields include economics and game theory (Mas-Colell, Whinston, & Green 1995), management science under the name decision analysis

Intended audience

3

(Keeney & Raiffa 1976), medicine (Weinstein et al. 1980) where utilities are often referred to as QALYs, and statistics (reviving the Bayesian approach; DeGroot 1970). Behavioral foundations ensure the intrinsic soundness of a decision model, preventing historical accidents such as happened for what is known as the separate-probabilitytransformation model (details in Chapter 5). Homeomorphic versus paramorphic modeling

A model is paramorphic if it describes the empirical phenomena of interest correctly, but the processes underlying the empirical phenomena are not matched by processes in the model (Harre´ 1970). For example, as emphasized by Milton Friedman (1953; see Bardsley et al. 2010 Box 2.4), market models can make correct predictions even if their assumptions about consumers do not match actual consumers’ behavior. A model is homeomorphic if not only its empirical phenomena match reality, but also its underlying processes do so. We will seek homeomorphic models of decision making. Not only do the decisions predicted by the model match the decisions observed, but also we want the theoretical parameters in the model to have plausible psychological interpretations. Friedman’s arguments in favor of paramorphic models are legitimate if all that is desired is to explain and predict a prespecified and limited domain of phenomena. It is, however, usually desirable if concepts are broadly applicable, also for future and as yet unforeseen developments in research. Homeomorphic models are best suited for this purpose. In recent years, economics has been opening up to introspective and neuro-imaging data. It is to be expected that the concepts of prospect theory, in view of their sound psychological basis, will be well suited for such future developments and for connections with other domains of research. Behavioral foundations with plausible preference conditions support the homeomorphism of a model. Intended audience

No particular mathematical background knowledge is required, besides a basic knowledge of probability theory and calculus. A willingness to work with formal models and to follow abstract trains of thought is needed for this book though. The measurement methods and behavioral foundations presented in this book will be as simple and transparent as possible, so as to be accessible to as many readers as possible. Mathematically sophisticated readers may be interested in this book, and will perhaps be surprised by it, from a didactic perspective. For example, Gilboa’s (1987) and Schmeidler’s (1989) rank-dependent utility theory, and Tversky & Kahneman’s (1992) new prospect theory have often been considered to be complex, with presentations based on a comonotonicity concept. These theories can, however, be presented and derived in an elementary manner if we use ranks instead of comonotonicity, as will be done in this book.

4

Introduction

Decisions under uncertainty are relevant in many fields, including finance, economics, psychology, management science, medicine, computer science, Bayesian statistics, and engineering. Readers from any of these fields can rest assured that no advanced concepts will appear from any of the other fields because the author does not have a bachelor’s degree in any of the fields mentioned. Attractive feature of decision theory

An attractive feature of decision theory is that the reader can always imagine that he or she is the decision maker. For each preference condition presented in the text, you can ask yourself whether you would want to satisfy this condition in your own decisions. It is easiest to read this book with this question in mind. Hence, the decision maker and the readers will usually be referred to as “you.” Structure

The material in this book has been structured so as to be accessible to readers with different backgrounds, levels, and interests. Many results that will be relevant to some readers but not to all have been stated in exercises, whose elaborations are in Appendix J. This structure gives different readers the chance to skip and select different parts. Italicized superscripts a, b, c indicate which exercises are suited for which readers. The superscript a refers to exercises that are easiest, and the superscript c refers to exercises that are most difficult and that will be of interest only to the most theoretically oriented readers. Many readers, especially empirically oriented readers who are not used to working with formal models, will want to skip almost all exercises. Typically, psychology students interested in formal models will be a-students who will study the empirical parts of this book; mathematical students are c-students who are required to study the theoretical parts; and economics students are somewhere in between, so that they are usually b students. The best way to completely master the material in this book – if there are no time restrictions – is to stop reading after every exercise and then first do that exercise. Readers who are satisfied with a less thorough and time-consuming study can use the exercises flexibly. Sometimes an exercise contains results that are needed to understand the rest of the text. This is indicated by an exclamation! as superscript. Then every reader, even those not doing the exercise, should read its results. Exercises are interspersed throughout the text, and are located where they are most relevant. Some sections conclude with assignments. These are further exercises that serve to grade students and/or to practice. Their results play no role in the theory development in the main text, and no elaborations of assignments are given in this book. On the author’s homepage, further exercises and assignments are provided. This serves teachers who wish to have more exercises without solutions available to the students. Teachers can obtain solutions to assignments from the publisher. Proofs of theorems are collected in appendices at the end of chapters.

Preview

5

For the use of this book, with comprehensive theoretical discussions and comprehensive discussions of empirical implications, Appendix K is instrumental. It illustrates how sections depend on preceding sections. By using this appendix, you need not read the book from start to finish. In a backward approach, you pick out any topic of interest, and then use Appendix K to see which preceding material you need to read for it. In a forward approach, you skip whatever you do not like. If needed later after all, then Appendix K will show you so. If you are interested in only part of the book, this organization allows you to use the book efficiently. In particular, teachers can easily select the material targeted at the interests of specific students. If you want to know the definition of prospect theory for unknown probabilities in }12.1, then you can select the texts depicted in Figure K.1 in Appendix K. The sections listed there comprise about 46 pages to be read. If you are not interested in the tradeoff technique of }4.1 and }4.5, then you can skip all of Chapter 4 except for }4.2 and }4.9, and then skip }}6.5, 9.4, 10.5, and 12.3. If you are interested only in decision under risk, then you can learn about the definition of prospect theory in }9.2, using the same method and the same figure, skipping virtually all sections on decision under uncertainty, and reading approximately 34 pages. If you want to learn about a pragmatic index of ambiguity aversion under prospect theory, then you can similarly use Figure K.2. If you want to understand as quickly as possible how the popular value at risk (VaR) for measuring the reliability of banks is a special case of prospect theory and rank dependence (Exercise 6.4.4), then you can find the shortest path: }}6.4, 6.3, 6.1, 3.2, 2.5, 2.4, 2.2, 2.1, 1.3, 1.2, 1.1. For 10 meetings of three hours each, a typical timetable may be: meeting 1: }1.1–}1.8; meeting 2: }2.1–}2.9; meeting 3: }3.1–}3.6, }4.1; meeting 4: }4.2–}4.7, }4.9.1, }4.11, }4.12; meeting 5: }5.1–}5.7, }6.1, }6.3–}6.5; meeting 6: }7.1–}7.4, }7.6–}7.11; meeting 7: }8.1–}8.5, }9.1–}9.5; meeting 8: }10.1–}10.6, }10.7.1, }10.8; meeting 9: }11.1, }11.4–}11.8; and meeting 10: }12.1–}12.3, }12.7. I have used this book in teaching advanced master’s students in economics who had digested large parts of Mas-Colell, Whinston, & Green (1995). I would then cover the material allocated to the first four meetings above in about two meetings, after which I would follow the above timetable. The total workload of this selection for students is about 120 hours of full-time work. A nice way to teach only part of this book is by restricting all models only to binary (two-outcome) prospects. This domain is rich enough to measure and define all components of risk attitude, utility, probability- or event-weighting, and loss aversion. Rank dependence and prospect theory are considerably simplified on this domain. This is how I taught this course to business students. They are particularly interested in prescriptive applications of decision theory. Preview

The book consists of three parts. Part I deals with the classical expected utility theory, and Parts II and III consider deviations. In Part I, }1.1 and }1.2 present the basics of decision under uncertainty. The rest of Chapter 1 presents the famous bookmaking

6

Introduction

condition of de Finetti, developed in the 1930s to justify the use of subjective probabilities for one-shot events. This condition is equivalent to the no-arbitrage condition in finance, which implies that market prices of financial derivatives have to be based on what are called as-if risk neutral evaluations. That is, these conditions imply expected utility when probabilities are unknown but utility is known (linear). Chapter 2 deals with expected utility when probabilities are known (“decision under risk”) but utilities are unknown. There are so many applications of this long-existing theory that they are presented separately in Chapter 3. Chapter 4 turns to the more complex topic of expected utility when both probabilities and utilities are unknown, using tradeoffs between outcomes as a tool to measure utility differences. It ends with some empirical violations of expected utility, preparing for the parts to follow. Part II deals with deviations from expected utility for decision under risk, where probabilities are known. We present rank-dependent utility, which generalizes expected utility by adding a new dimension of risk attitude: probabilistic sensitivity – i.e., the nonlinear ways in which people may process probabilities. This dimension is descriptively as relevant for risk attitudes as the nonlinear ways in which people process outcomes (utility), and had been sorely missing in the models used before. In 1982 John Quiggin introduced a correct theoretical manner of modeling such nonlinear processing, the rank-dependent formula. It was only then that a serious descriptive analysis of risk attitudes could begin. Chapter 5 presents mathematical and psychological arguments to show that the rank-dependent model naturally captures probabilistic sensitivity. Chapter 6 defines the theory formally, and shows how it can be used to tractably capture prevalent phenomena regarding risk attitude. We use ranks, introduced by Abdellaoui & Wakker (2005), as a tool to measure probability weight differences. We can then define ranked probabilities, which are the analogs in the probability dimension of the tradeoffs in the outcome dimension used in Chapter 4. Ranked probabilities facilitate the analyses of the rank-dependent model and are more tractable than the comonotonicity concepts that have been used in the literature. Chapter 7 presents empirical findings and special cases of rank dependence. In Chapters 8 and 9 we turn to prospect theory. In 1992, Tversky and Kahneman incorporated Quiggin’s idea of rank dependence to solve a theoretical problem of their original prospect theory of 1979. It led to the present version of prospect theory, also called cumulative prospect theory. To prepare for prospect theory, Chapter 8 introduces another generalization of expected utility beyond rank dependence: reference dependence. Outcomes are reinterpreted as changes with respect to a reference point (often the status quo). With reference dependence introduced, all ingredients are now available to define and analyze prospect theory for risk (Chapter 9). Part III concerns decision under uncertainty, where probabilities need not be known. Ambiguity attitudes, which deviate from expected utility in fundamental ways and may not even admit the existence of (subjective) probabilities, are analyzed. Chapter 10 starts by extending Quiggin’s definition of rank dependence from risk to the more subtle context of uncertainty, for which Schmeidler (1989, first version

Our five-step presentation of decision models

7

1982) conceived it independently. Chapter 11 presents the main novelties of uncertainty, namely source dependence, which includes ambiguity aversion. We show how rank dependence can be used to analyze uncertainty, and to provide tractable measures of ambiguity aversion and sensitivity to ambiguity. These measures encompass the currently popular a-maxmin model. Chapter 12 presents the most important model of this book, namely prospect theory for uncertainty. This model entails a common generalization of all the models presented up till then. Relative to Chapters 10 and 11, it allows ambiguity attitudes for losses to be different than for gains. This generalization is desirable because empirical studies have shown that such differences are pronounced. Prospect theory is the first theory for decision under uncertainty that is both theoretically sound and empirically realistic. It means that only since 1992 do we have a satisfactory theory that can deliver the full spectrum of what is needed for decision under risk and uncertainty. Chapter 13 concludes the main text. Appendices A–K complete the book. I will only discuss a few here. }A.1 in Appendix A contains a general methodological discussion of models being imperfect with inconsistencies in data, and of the nonparametric measurements that are central to this book. Appendix B presents some general issues of the revealed-preference paradigm. Appendix F shows that the influential Fehr–Schmidt model for welfare evaluations is a special case of rank dependent utility, and Appendix J contains the elaborations of the exercises in the book. Our five-step presentation of decision models

We usually present decision theories in five steps that serve to make the empirical meaning of the theories tangible for the readers. The first step is, simply, to define the decision model. We specify the subjective parameters of the model and give a formula describing how these parameters imply preferences. In expected utility with given probabilities (risk), the subjective parameter is utility. In prospect theory for risk, the subjective parameters are utility, probability weighting, and loss aversion. In the second step, it is demonstrated how decisions can be derived from the model using simple numerical exercises. Although we do not endorse Lord Kelvin’s maxim “science is measurement” as a universal principle, measurement is central to this book. Thus, the third step in our analysis presents what is called the elicitation method. It demonstrates how the subjective parameters of a decision theory can be measured from observed preferences in as simple and direct a way as possible. The third step reverses the second step. Now, preferences are not derived from the subjective parameters but the subjective parameters are derived from preferences. To illustrate the third step for readers who know the expected utility model, assume expected utility with utility U for given probabilities. Assume a scaling U($0) ¼ 0 and U($100) ¼ 1 (such scaling is always possible as we will see later). Then, for any $a between $0 and $100, an indifference between receiving $a for sure or receiving

8

Introduction

$100 with probability p and receiving $0 with probability 1p immediately implies U($a) ¼ p, as we will see later. In this manner we can relate utility to observed choice in a direct and simple manner. Using such observations, we always obtain the right utility function whatever this function is. For instance, we need not assume that utility belongs to a particular family such as the family of power functions (“CRRA”; U(a) ¼ ay for some power y). In descriptive applications we can use such measurements of utility to make predictions about future preferences. In prescriptive applications, simple indifferences as just described can be used to determine preferences in real and complex situations. A necessary condition for a model to hold is that different ways to measure its subjective parameters should not run into contradictions. This leads to the fourth step in our analysis of decision models: we define preference conditions to preclude such contradictions. These preference conditions then turn out to be not only necessary, but also sufficient, for the decision model considered to hold. That is, they provide behavioral foundations of the models, presented in the fifth step. This means that the behavioral foundations in this book will all be closely related to direct empirical measurements of the subjective parameters of models. The preference conditions are thus easily observable and testable.1 They all involve a limited number of tractable choices. Comment 2.6.4 will describe the five steps for the expected utility model in Chapter 2. I first developed these five steps when I taught prospect theory to a selected group of high school students (in 30 hours), where this approach worked well. I have maintained it ever since. After having thus introduced decision models, we discuss their empirical properties, and give suggestions for applications to various fields. I hope that this book will enhance further applications, to be developed by readers specialized in such fields.

1

Gilboa (2009 end of Ch. 8) pointed out that, conversely, simple preference axioms (¼ conditions) can be used to obtain simple elicitation methods: “Yet, the language of the axioms, which is concrete and observable, and the quest for simple axioms often facilitate the elicitation/calibration problem significantly.”

Part I Expected utility

1

1.1

The general model of decision under uncertainty and no-arbitrage (expected utility with known utilities and unknown probabilities) Basic definitions

This section introduces the general notation and terminology for decision under uncertainty. Examples of decision making under uncertainty Example 1.1.1 [Vendor]. Imagine that you are a street vendor. You have to decide which merchandise to take along with you tomorrow. Merchandise can be ice cream, hot dogs, newspapers, magazines, and so on. The net profits resulting from the merchandise depend on the weather tomorrow, and can be negative because goods not sold tomorrow are lost. Although weather conditions constitute a continuum of possibilities, we assume for simplicity that we need to distinguish only between three eventualities: either there will be no rain (s1), or there will be some rain (s2), or it will rain all day (s3). You do not know for sure whether s1, s2, or s3 will occur. Table 1.1.1 presents, in dollar units, the profits that can result from your decision depending on the weather tomorrow. Table 1.1.1. Net profits obtained from merchandise, depending on the weather

x (“ice cream”) y (“hot dogs”) 0 (“neither”) x þ y (“both”)

no rain (s1)

some rain (s2)

all rain (s3)

400 400 0 0

100 100 0 200

400 400 0 0

We assume, for simplicity, that the supply of ice cream does not affect the profits obtained from hot dogs, and vice versa, in the lowest row “both.” □ Example 1.1.2 [Finance]. Imagine that you want to speculate on the copper price next month. You can buy a financial portfolio, denoted x and costing 30K (K ¼ $1000), 11

12

Chapter 1: Dutch books/No-arbitrage

that yields 80K in the event E1 of the copper price (per pound) exceeding $2.53 next month, and nothing otherwise. Such portfolios can be replicated by proper combinations of options and other financial instruments (Hull 2005 end of }10.4). All payments take place next month, so that we can ignore discounting. The net profits are shown in Table 1.1.2. You can also buy a portfolio y at the same price of 30K, yielding 80K in the event E3 of the copper price falling below $2.47 and nothing otherwise. E2 denotes the residual event of a copper price being between $2.47 and $2.53. You can also buy both portfolios, or buy neither. Table 1.1.2. Net profits depending on the copper price

x y 0 (“neither”) x þ y (“both”)

price  2.53 (E1)

2.53 > price  2.47 (E2)

2.47 > price (E3)

50K 30K 0 20K

30K 30K 0 60K

30K 50K 0 20K

□ Example 1.1.3 [Elections]. You have to decide which investment to choose in a foreign country. There will be elections in this foreign country, and exactly one of the participating candidates will win. sj designates the eventuality of the jth candidate winning. The net profits of an investment depend on which candidate will win. □ Example 1.1.4 [Horse race]. You are at a horse race, and have to decide which gamble to play. Exactly one horse will win. sj designates the eventuality of horse j winning. Not gambling is one of the “gambles” available to you, yielding outcome 0 whichever horse will win. □ Example 1.1.5 [Medical example]. You are a doctor and have to decide on the treatment of a patient. The patient has exactly one of a number of potential diseases, and it is uncertain which one it is. The outcome of the treatment – i.e., the health state resulting for the patient – depends on the treatment chosen and on the disease of the patient. □ Example 1.1.6 [Agriculture]. You are a farmer and have to decide which crops to grow. The resulting profits will depend on the crops chosen and on the weather conditions during the coming year, which are uncertain. □ Many other didactical examples are in Lavalle (1978 }1.5) and throughout Clemen (1991). The basic concepts of uncertainty. We will model uncertainty through a state space S. One state s ∊ S is true and the other states are not true. It is uncertain which state is the true one. The decision maker does not have any influence on which state is true (for details, see }1.7). In Example 1.1.1 (vendor), the uncertainty concerns the weather, and the state space S is {s1,s2,s3}. In Example 1.1.2 (finance) the uncertainty

1.1 Basic definitions

13

concerns the copper price next month. We take the nonnegative real axis as state space, so that S ¼ [0,1) and S is infinite. In Example 1.1.4, the uncertainty concerns which horse will win and S is the set of participating horses. Every horse corresponds to the state that this horse will win the race. States are also called states of nature. Although the terms states and states of nature may not be the most natural ones, they are generally used in decision theory and we will use them too. Subsets of the state space are called events. In Example 1.1.2, the subset [0, 2.47) of S concerns the event of the copper price falling below $2.47; i.e., it is event E3. If, in Example 1.1.4, horses s1, s3, s6 are French horses, and the others are not, then the event “A French horse will win” is described by the subset {s1,s3,s6} of S. An event is true if it contains the true state. Event {s1,s3,s6} is true if the true state is s1, s3, or s6, that is, if a French horse wins. For event E, Ec denotes the complementary event SE of all states s ∊ S not contained in E; it is the negation of E. The outcome set is Ñ, with real numbers designating amounts of money. Apart from a few exceptions specified later, this outcome set is assumed throughout this book. Thus, we mostly deal with financial risks. Outcomes are usually denoted by Greek letters a, b, g, d, or by Roman letters with subscripts such as x1. In this book, subscripts virtually always refer to events, and superscripts are used for all other indexing purposes. Superscripts thus usually do not refer to power-taking. We use the term increasing in the same, weak, sense as nondecreasing: A function U: Ñ ! Ñ is increasing, or nondecreasing, if a > b implies U(a)  U(b). It is strictly increasing if a > b implies U(a) > U(b). Decreasing, nonincreasing, and strictly decreasing are defined similarly. Prospects designate state-contingent outcomes. They are courses of action, the outcome of which depends on which state of nature is the true one. Formally, prospects map states to the reals, describing the resulting outcome for every state if that state is the true state. They are random variables, but possibly without probabilities specified. We assume throughout that prospects take only finitely many values. In the vendor Example 1.1.1, prospects describe the net profits (which can be negative in the case of a loss, such as for unsold ice cream that will be lost) conditional on each weather condition. Prospect x assigns x(s1) ¼ 400 to s1, x(s2) ¼ 100 to s2, and x(s3) ¼ 400 to s3. In the finance Example 1.1.2, prospects describe the net profits conditional on the copper price next month. Notation for prospects. Prospects x are often denoted as (E1:x1, . . ., En:xn), yielding x1 under event E1 (i.e. for each state s ∊ E1), . . ., and xn under event En. In the finance Example 1.1.2, x can be written as (E1:50K, E2:30K, E3:30K). In the vendor Example 1.1.1, x can be written as (s1:400, s2:100, s3:400). For x þ y in Example 1.1.1 we can write E1 ¼ {s1,s3}, E2 ¼ s2, and then we have x þ y ¼ (E1:0, E2:200). We often suppress brackets, colons, and commas, if no confusion will arise. We may thus write E1x1  Enxn or E1x1,E2x2,E3x3. If there are only two outcomes and events, such as in (E:a, Ec:b), then we often suppress the second event, and write aEb. For example, for x þ y in the vendor Example 1.1.1 we may write 0E1200 rather than (E1:0, E2:200). In the finance

14

Chapter 1: Dutch books/No-arbitrage

Example 1.1.2, for x þ y we may write 60KE220K. In the agriculture Example 1.1.6, 2000E12000 ¼ (E1:2000, E1c:2000) may refer to the growing of a crop that will  die, bringing a net loss of 2000, if the minimum temperature is below 32 F (E1), and that will survive otherwise, bringing a net profit of 2000 (E2 ¼ E1c). Because it is uncertain which state is true, it is uncertain what the outcome of a chosen prospect will be, and the decisions indeed have to be made under uncertainty. The prospects Aa,Ba,Cb and (A [ B)a,Cb assign the same outcome to each state of nature, so that they refer to the same map from S to Ñ. Given that prospects are defined to be such maps, Aa,Ba,Cb and (A [ B)a,Cb are identical by definition. For different formal definitions where different ways of notation (“framing”) can matter, see Birnbaum (2008a) and Luce (1990, 2000). Consider, in the vendor Example 1.1.1, the prospect (s1:100, s2:100, s3:100). It yields outcome 100 for sure, irrespective of the weather conditions. Such prospects, yielding the same outcome for each state, are called constant prospects. For a constant prospect there is no uncertainty about the outcome, and we often denote the prospect simply as that outcome. For example, we write 100 instead of (s1:100, s2:100, s3:100). In the finance Example 1.1.2, the prospect denoted 0 is constant, yielding 0 profit whatever the copper price. For a given prospect denoted E1x1  Enxn, the events Ej are called the outcome events. These are the payoff-relevant events. It is implicitly assumed for each prospect that the outcome events are exhaustive and mutually exclusive. That is, they partition the state space S. Exactly one of them is true (contains the true state) and the others are not true. If the partition (E1,. . .,En) is understood, we sometimes suppress the events and write, briefly, (x1,. . .,xn). Such partitions will be central to this book, and the state space S will usually be in the background. We sometimes use the longer term event-contingent prospect instead of prospect, to distinguish the general prospects defined here from their subclass called (probability-contingent) prospects that will be defined in later chapters. The domain of preference. We assume throughout that all maps from S to Ñ taking finitely many values are prospects, and are available in the domain of preference. Preferences and evaluations. The preference relation of the decision maker, usually assumed to be you, over the prospects is denoted by ≽, a curved version of the greater-equal sign . x ≽ y means that you prefer x to y and are willing to choose x from {x,y}. We thus interpret preference as nothing other than binary choice. Multiple choice can be derived from binary choice by assuming that from the multiple available prospects a prospect is chosen that is best according to binary preference (Appendix B). We use the following notation for preferences. strict preference: x  y if x ≽ y and not y ≽ x; equivalence/indifference: x  y if x ≽ y and y ≽ x; reversed preference: x ≼ y if y ≽ x; strict reversed preference: x ≺ y if y  x. Hence, if you prefer x to y (x ≽ y) and are willing to choose x from {x,y}, then either you consider x to be equally preferable to y (x  y) and you would also be willing to

1.2 Basic properties

15

choose y, or you consider x to be strictly preferable (x  y) and you are not willing to choose y. A certainty equivalent (CE) of a prospect x is an outcome a such that a  x where, as often, the symbol a also designates a constant prospect. If you just received x, its CE will be the minimum price at which you are willing to sell x. A function V represents ≽ if V evaluates prospects in agreement with preference. That is, V assigns to each prospect a real number that indicates the value of the prospect in the sense of the following logical equivalence: [x ≽ y , V(x)  V(y)]. That is, x ≽ y if and only if V(x)  V(y). We then also say that ≽ maximizes V. Examples of representing functions are the expected value function examined later in this chapter, and the expected utility function examined in Chapter 2. The following two exercises serve to clarify the basic model of decision under uncertainty. Exercise 1.1.1.a Assume that S ¼ {Bill, Jane, Kate, no-one}, describing who will first enter your office before lunch, if anyone does. In deviation from the default assumption in this book, do not assume that the outcome space is Ñ, but assume that it contains only two elements, a (apple) and n (nothing), designating what you will receive for food before lunch. (a) Describe formally the events A: a man will enter first; B: no woman will enter first; C: someone will enter; D: a woman will enter first; E: someone whose name contains an “a” will enter first. (b) Use the formal prospect-notation of this section to denote the following prospects: x. You receive an apple if and only if no man enters; y. You receive an apple if and only if Kate enters. z. You receive nothing if and only if a woman or no-one enters. (c) How many prospects do there exist? (d) How many constant prospects do there exist? Write them using the notation of this section. □ Exercise 1.1.2.b Assume that probabilities are given in Example 1.1.4, with P(s1) ¼ ½, P(s2) ¼ ¼, and P(s3) ¼ ¼. Two events A and B are (stochastically) independent if P(A\B) ¼ P(A)  P(B). Are s1 and s2 independent? □ 1.2

Basic properties of preferences

Whereas no theoretical result in this book can be understood completely without having studied this section, this section may still be skipped by readers with a minimal theoretical interest. If a representing function exists, then ≽ is a weak order; i.e., ≽ is transitive (if x ≽ y and y ≽ z then x ≽ z) and complete (for all x, y, x ≽ y or y ≽ x, where possibly both preferences hold). Weak ordering entails a ranking of prospects with ties permitted. A violation of transitivity suggests a violation of a conservation-of-preference principle. With x ≺ z, z ≼ y, and y ≼ x, we could create

16

Chapter 1: Dutch books/No-arbitrage

positive preference value through each replacement [x ! z], [z ! y], [y ! x], yet ending up in a final position that has not been improved relative to the initial position. Bear in mind in the following exercise that an exclamation mark indicates that it is useful for all readers to read the results of the exercise. Exercise 1.2.1.!a Show that ≽ is a weak order if a representing function V exists. □ Exercise 1.2.2.!a Show that, if V represents ≽, then: (a) [x  y , V(x) > V(y)]. (b) [x  y , V(x) ¼ V(y)]. □ Here are further properties of preference relations ≽ that will usually, although not always, be satisfied in what follows. Nontriviality: x  y for some prospects x,y. Reflexivity: For all x, x  x. Monotonicity: if x(s)  y(s) for all states s, then x ≽ y, and, if x(s) > y(s) for all states s, then x  y. It is plausible that your preference relation – say, over bets on a horse race – satisfies all these conditions. The conditions in the following exercise are also natural. Exercise 1.2.3.!c Assume that ≽ is a weak order. Show that: ≽ and  are reflexive.  is transitive. For no x and y we have x  y and y  x ( is asymmetric). For no x and y we have x ≽ y and y  x. If x ≽ y and y  z then x  z. If x  y and y ≽ z then x  z. If x ≽ y, y ≽ f, f ≽ g, then x ≽ g.  is an equivalence relation; i.e.,  is reflexive, transitive, and symmetric (x  y then y  x). (i) If y  x then y is substitutable for x in every preference; i.e., [x ≽ f , y ≽ f], [x ≼ f , y ≼ f], [x  f , y  f], [x ≺ f , y ≺ f], and [x  f , y  f]. (j) x  y , (not y ≽ x). □

(a) (b) (c) (d) (e) (f) (g) (h)

The result of the following exercise is often useful. It implies that we may assume in many situations that all prospects considered refer to the same partition E1,. . .,En. Exercise 1.2.4b [Partition flexibility]. (a) Consider two prospects x ¼ A1a1  Akak and y ¼ B1 b1    B‘ b‘ . Show that there exists one partition E1,. . .,En such that x ¼ E1x1  Enxn and y ¼ E1y1  Enyn for properly chosen x1,. . .,xn, y1,. . .,yn, with n ¼ k‘. Note that, if Ai is logically incompatible with Bj, then an event Ek referring to their intersection will be the empty set, referring to the vacuous, impossible, event. It is permitted that some events Ek are empty.

1.3 Expected value

17

(b) Explain in spoken words, without writing, that for every finite set of prospects there exists one partition E1,. . .,En such that all prospects considered are of the form E1x1  Enxn. □ Exercise 1.2.5.b (a) Assume monotonicity and weak ordering. Show that [a ≽ b , a  b]. (b) Assume monotonicity, weak ordering, and assume that for each prospect x a certainty equivalent CE(x) exists. Show that CE represents ≽. □ To avoid trivialities, we assume nondegeneracy: gEg  gEb  bEb for some event E and outcomes g,b. For example, assume 100no rain100  100no rain0  0no rain0 in the vendor Example 1.1.1. The last, second, preference shows that the event of no rain can affect preference and should be reckoned with, and the first preference shows the same for the complementary event. Nondegeneracy ensures that there is real uncertainty and that you do not know the true state of nature for sure, because you apparently reckon with the possibility of E containing the true state but also with the possibility of Ec containing the true state. We summarize the assumptions made. Structural Assumption 1.2.1 [Decision under uncertainty]. S is a, finite or infinite, state space, and Ñ is the outcome set. Prospects map states to outcomes, taking only finitely many values. The domain of preference is the set of all prospects, i.e., of all such maps. ≽ is a preference relation on the set of prospects. Nondegeneracy holds. □ Assignment 1.2.6.b Show that, under weak ordering: If x ≽ y, y ≽ f, f ≽ g, g  h, then x  h. □ 1.3

Expected value

One plausible way to make decisions under uncertainty is by choosing probabilities, sometimes called subjective, for the uncertain events, and then maximizing expected value with respect to those probabilities. Probabilities P satisfy, besides the obvious P(S) ¼ 1 and P() ¼ 0, additivity: P(A [ B) ¼ P(A) þ P(B) for all disjoint (empty intersection; incompatible) events A and B. Definition 1.3.1. Expected value (EV) holds if there exist (subjective) probabilities P such that preferences maximize the expected value (EV) of prospects defined by E1 x1    En xn 7! PðE1 Þx1 þ    þ PðEn Þxn :

h

Under EV, the (subjective) probabilities are the subjective parameters that characterize the decision maker. The following exercise shows how to derive decisions from the EV model. Its elaboration is immediately given in the main text. Exercise 1.3.1.a Consider the vendor Example 1.1.1. Assume EV with P(no rain) ¼ 0.40, P(some rain) ¼ 0.30, and P(all rain) ¼ 0.30. Calculate the EV of the four prospects in Table 1.1.1, and determine which prospect is chosen. □

18

Chapter 1: Dutch books/No-arbitrage

In Exercise 1.3.1: EV(x) ¼ 0.40  400 þ 0.30  100 þ 0.30  (400) ¼ 70; EV(y) ¼ 0.40  (400) þ 0.30  100 þ 0.30  400 ¼ 10; EV(0) ¼ 0; EV(xþy) ¼ 0.30  200 ¼ 60. Hence x has the highest EV and is chosen. Exercise 1.3.2.a Consider the vendor Example 1.1.1. Assume EV with P(s1) ¼ ½, P(s2) ¼ ¼, and P(s3) ¼ ¼. What is the preference between x ¼ 300s10 and y ¼ 200s3100? □ In the literature, the term subjective expected value has often been used to stress that the probabilities need not have an objective basis but may be subjective. Market probabilities in finance are an example of subjective probabilities. Throughout this book we will treat objective probabilities (risk) not as a different case, but as a subcase, of subjective probabilities (uncertainty). It is, therefore, convenient to use the same terms for risk and for uncertainty whenever possible, which is why we usually suppress “subjective.” EV can be used to derive preferences as we just saw. We now discuss the elicitation method, which consists of reversing the direction of derivation. For example, if from EV(x) > EV(y) we can derive x  y, then from an observation y ≽ x we can derive EV(y)  EV(x). In general, from observed preferences we can derive properties of decision models. Such reversed moves, where empirical observations are used to derive implications for a theoretical model, are indispensable for the operationalization of every empirical field. Thus, the derivation of utility from preference has a long tradition in economics (Stigler 1950). De Finetti (1931a) and Ramsey (1931) provided remarkable early examples in decision theory. They showed how the subtle concept of subjective probability can be derived from observed preferences. Another remarkable example is Samuelson (1938) for another context (Appendix B)1. The reversed move described in this paper played an important role in the ordinal revolution in economics (Lewin 1996). William James (1884, p. 190) wrote, on a similar reversed move for the study of emotions: “that the more rational statement is that we feel sorry because we cry, angry because we strike, afraid because we tremble, and not that we cry, strike, or tremble, because we are sorry, angry, or fearful.” Exercise 1.3.3.a Assume, in Example 1.1.1, that we observe 400s10  400s30. Show that, under EV, P(s1) > P(s3). □ In a descriptive context, if we know that EV holds but we do not know the subjective probabilities of a decision maker, then we can derive those from observed preferences. This is illustrated by the following exercise, which is particularly useful. If you have no clue how to resolve it, then you probably have not yet fully internalized the conceptual meaning of decision models. Then a rereading of some of the preceding text is useful – rather than reading the solution to the exercise, which is given in the text immediately following the exercise. 1

Because of the difference in context, we cannot use Samuelson’s term “revealed preference” for the elicitation method.

1.4 Data fitting for EV

19

Exercise 1.3.4.a Assume that you want to find out, in Example 1.1.1, what the probabilities P(s1), P(s2), and P(s3), denoted p1, p2, p3, are of the street vendor. However, the only thing that you can observe about the street vendor is his preferences between prospects. That is, for each pair of prospects x, y, you know which of the two the street vendor prefers and, for instance, for each prospect you know the certainty equivalent. This holds for all prospects, that is, for all mappings from the states to the reals and not just for those considered in Example 1.1.1. You know that the street vendor maximizes expected value. How can you find out the street vendor’s probabilities p1, p2, p3? In other words, how can you elicit these probabilities from the street vendor’s preferences? (What is the easiest way you can think of?) □ If expected value holds, then the probabilities can easily be elicited from preferences as follows. For an event E, find the number 0 p 1 such that p  1E 0:

ð1:3:1Þ

Then p ¼ PðEÞ  1 þ ð1  PðEÞÞ  0 ¼ PðEÞ:

ð1:3:2Þ

Eq. (1.3.1) shows that p can be interpreted as an index of your willingness to bet on event E. The equations give a direct link between subjective probabilities and preferences. Probabilities inferred from choice this way can be used to predict future decisions. Exercise 1.3.5.a Assume EV in the vendor Example 1.1.1 with 100s10  50 and 100s20  25. What are P(s1), P(s2), P(s3), and what is CE(100s30)? What is the preference between 100s20 and 100s30? □ In a general normative context on decision under uncertainty, if we have made up our mind on a number of preferences (such as the first two in Exercise 1.3.5), then these can be used to determine probabilities from which subsequently other preferences can be derived. Such types of inferences are typical of decision theory as presented in this book. 1.4

Data fitting for expected value

This section gives numerical examples and exercises to illustrate how EV can be used to describe and predict preferences. It can be skipped by readers not interested in data fitting (as can be inferred from the figures in Appendix K). Interested readers have to read }A.2 in Appendix A. Throughout this section we assume the vendor Example 1.1.1. The perspective will be descriptive. We assume that the street vendor is someone else and you are observing and predicting the street vendor’s choices without a commitment to whether or not these choices are rational. Assumption 1.4.1. Three states of nature (events) s1, s2, and s3 are given. These events are suppressed in the notation of prospects, as in (x1,x2,x3). EV holds with, possibly unknown, probabilities P(s1) ¼ p1, P(s2) ¼ p2, and P(s3) ¼ p3. □

20

Chapter 1: Dutch books/No-arbitrage

The solution to the following exercise is immediately given in the ensuing example. Exercise 1.4.1a [Deterministic nonparametric measurement]. We observe the following indifferences: (4,0,0)  2, (0,3,0)  1, and (3,0,3)  2. (a) Show that EV can accommodate these observations, and determine the probabilities of the three events. (b) Predict the CEs of the prospects (6,6,12) and (16,0,0), and predict the preference between these prospects, assuming EV. □ Example 1.4.2 [Elaboration of Exercise 1.4.1]. Assume the indifferences of Exercise 1.4.1. We consider the first two and apply EV calculation. 4p1 ¼ 2 implies p1 ¼ 1/2 and 3p2 ¼ 1 implies p2 ¼ 1/3 . Then p3 must be 1  1/2  1/3 ¼ 1/6 . The third indifference is consistent with these results with, indeed, ð1/2 Þ  3 þ ð1/6 Þ  3 ¼ 2. We then have EVð6; 6; 12Þ ¼ 6/2 þ 6/3 þ 12/6 ¼ 7 and, hence, CEð6; 6; 12Þ ¼ 7: EVð16; 0; 0Þ ¼ 16/2 ¼ 8 and, hence, CE(16,0,0) ¼ 8. The resulting preference is (6,6,12) ≺ (16,0,0). □ The probabilities in Example 1.4.2 could be obtained particularly easily because the data came in a form similar to Eq. (1.3.1). In general, the derivation of subjective parameters such as probabilities from data may be less easy and usually it requires resolving linear equations if indifferences are given or it requires solving linear inequalities if preferences are given. For example, indifferences (4,12,6)  7 and (4,6,6)  (8,3,0) can be seen to imply p1 ¼ 1/2 , p2 ¼ 1/3 , and p3 ¼ 1/6 , as in Example 1.4.2, but the derivation is less easy.2 In this book we will strive for measurements, and preference conditions derived from those, that are directly targeted towards the parameters of interest, as in Example 1.4.2. That is, the data have been devised such that the subjective parameters, such as subjective probabilities, can be derived from the data as easily as possible. Thus, the subjective parameters are linked to empirical choices in a clear and transparent manner. In the following example EV does not hold exactly, but is used to approximate observed preferences as closely as possible. Example 1.4.3 [Measurement with errors and fitting]. We observe the following indifferences: (4,0,0)  2, (0,3,0)  1, and (3,0,3)  1.5. EV fails as readily follows from Example 1.4.2: The first two indifferences imply p1 ¼ 1/2 and p2 ¼ 1/3 , so that p3 ¼ 1/6 , but then CE(3,0,3) should be 2 and not 1.5 as it is here. We may nevertheless use EV to approximate the data. Then either we assume that the observed data contain errors, or we are satisfied with an approximate model. We may find the best-fitting probabilities as follows, using a special case of a general distance measure presented in }A.2. We consider each possible probability vector (p1,p2,p3) of the form (i/100, j/100, (100ij)/100), with i and j integers. For each we consider by how far the CEs of the 2

The equations implied are 4p1 þ 12p2 þ 6(1p1p2) ¼ 7 and 4p1 þ 6p2 þ 6(1p1p2) ¼ 8p1 þ 3p2. These have the solution mentioned.

1.4 Data fitting for EV

21

three prospects predicted by the probability vectors are off from the observed CEs. For instance, for the first prospect the predicted CE is 4p1, so that the probability vector is off by |4p1  2|, with squared distance (4p1  2)2. We similarly take the squared distances for the other two prospects, take the average of these three squared distances, and then take its square root. It leads to the following distance formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4p1  2Þ2 þ ð3p2  1Þ2 þ ð1:5  3p1  3p3 Þ2 : 3 The distance has money as its unit and can be interpreted as the amount by which the model is off in dollars. By calculations for various combinations of probabilities using a computer3 it can be verified that the probabilities p1 ¼ 0.50, p2 ¼ 0.42, and p3 ¼ 0.08 best fit the data. They give certainty equivalents CE(4,0,0) ¼ 2, CE(0,3,0) ¼ 1.26, and CE(3,0,3) ¼ 1.74, with distance sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2  2Þ2 þ ð1:26  1Þ2 þ ð1:74  1:5Þ2 ¼ 0:20: 3 To predict the preference between (6,6,12) and (16,0,0), we find that the CEs for these prospects are 6.48 and 8, respectively, so that the predicted preference is (6,6,12) ≺ (16,0,0). □ Exercise 1.4.2a [Deterministic nonparametric measurement]. Assume that we observe the following indifferences: (8,0,0)  2, (8,8,0)  4, and (0,4,4)  3. (a) Show that EV can accommodate these observations, and determine the probabilities of the three events. (b) Determine the CEs for the prospects (6,6,12) and (32,0,0), and the preference between these prospects. □ Exercise 1.4.3b [Measurement with errors and fitting]. We observe the following indifferences: (8,0,0)  2, (8,8,0)  4, and (0,4,4)  2. (a) Show that EV cannot accommodate these observations. (b) Take a distance measure similar to Example 1.4.3, which now amounts to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8p1  2Þ2 þ ð8p1 þ 8p2  4Þ2 þ ð4p2 þ 4p3  2Þ2 : 3 For each of the following three probability vectors (p1,p2,p3) determine their distance from the data,4 and determine which of them fits the data best: (0.25, 0.25, 0.50), (0.28, 0.22, 0.50), and (0.24, 0.24, 0.52).

3 4

In this case, an exact analytical method using first-order optimality conditions can easily be obtained. In many later exercises, no analytical solution can be obtained and numerical analyses using the computer have to be used. We equate a probability tuple with its corresponding EV model in what follows.

22

Chapter 1: Dutch books/No-arbitrage

(c) Which probability vector (i/100, j/100, (100ij)/100) fits the data best? What is its distance? Predict the CE’s of (8,0,8) and (24,0,0), and the preference between these prospects. □ Assignment 1.4.4a [Measurement with errors and fitting]. We observe the following indifferences: (10,0,0)  1, (0,10,0)  2, and (10,10,0)  3. (a) Show that EV can accommodate these observations, and determine the probabilities of the three events. (b) Predict the CE’s of (7,7,0) and (0,0,3), and the preference between these prospects. □ Assignment 1.4.5a [Measurement with errors and fitting]. We observe the following indifferences: (10,0,0)  1, (0,10,0)  2, and (10,10,0)  4. (a) Show that EV cannot accommodate these observations. (b) Take a distance measure similar to Example 1.4.3, which now amounts to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10p1  1Þ2 þ ð10p2  2Þ2 þ ð10p1 þ 10p2  4Þ2 : 3 Determine the distance for each of the following three probability vectors, and determine which of them fits the data best: (0.10, 0.20, 0.70), (0.12, 0.22, 0.66), and (0.08, 0.18, 0.74). (c) Which probability vector (i/100, j/100, (100ij)/100) best fits the data? What is its distance? Predict the CE’s of (4,0,4) and (24,0,0), and the preference between these prospects. □ 1.5

The bookmaking argument of de Finetti, or the no-arbitrage condition from finance

Whereas the preceding section primarily concerned descriptive applications of expected value, using it to explain and predict observed data, the analysis in the next two sections primarily concerns prescriptive and normative applications. In Example 1.1.1, imagine that you are the street vendor and want to make decisions as wisely as possible. Alternatively, the street vendor could be someone else, to whom you want to give the best advice. In Eqs. (1.3.1) and (1.3.2) we demonstrated how subjective probabilities are directly related to preferences. A behavioral foundation of expected value can be obtained by imposing consistency on such measurements. This is the topic of Assignment 1.6.9. We will use a somewhat different method in this section and the next one to obtain a behavioral foundation. In finance, arbitrage means that you can combine a number of portfolios in such a way that you always (in every state of the world) make a profit. In such a case, you can make money from the market without contributing anything, which you will of

1.5 de Finetti’s Dutch books

23

course do as much as possible. In practice, arbitrage opportunities disappear within fractions of seconds because they are immediately exploited by the largest and quickest market participants, i.e. investment banks that use automated computer programs. A fundamental result in finance is that no-arbitrage implies an as-if risk neutral (¼ EV) evaluation of financial assets. This result will be proved in the next section. The same result was discovered before by de Finetti (1931a) for individual choice, where arbitrage is called a Dutch book. De Finetti’s bookmaking result has been fundamental in decision theory because it provided the basis for subjective probability and Bayesian statistics. De Finetti (1931a) formulated his idea as a game between a bookmaker and a bettor where it was in the interest of the bookmaker to report true subjective probabilities. This was probably the first incentive-compatible mechanism presented in the literature, preceding the economics Nobel prize winner Hurwicz (1972). In finance, the market sets prices for all financial portfolios and they can be bought and sold in any desired quantity at that price. We assume throughout that there are no transaction costs. To apply our techniques of individual choice to the market, we model the market as one decision maker. The price of a portfolio (¼ prospect) is interpreted as its certainty equivalent. Preferences between portfolios are represented by the certainty equivalents. The ease with which we can restate the fundamental noarbitrage theorem from finance as a result from decision theory illustrates the flexibility of the decision-theory model (summarized in Structural Assumption 1.2.1). Prospects can be added, as for instance in the last row of the tables in Examples 1.1.1 (vendor) and 1.1.2 (finance) for prospects x, y. The prospect x þ y assigns x(s) þ y(s) to each state s. The following condition is central to this and the following section. Definition 1.5.1. Additivity holds if: [x ≽ y ) x þ z ≽ y þ z] for all prospects x,y,z. □ That is, improving an ingredient in a sum improves the sum. The condition is selfevident in finance, where, if buying x is more expensive than buying y, then buying both x and z is obviously more expensive than buying both y and z. Additivity is illustrated in Tables 1.5.1–1.5.4. In Table 1.5.1, prospect z is constant, and its outcome does not depend on the weather tomorrow. In Table 1.5.2, the outcome of prospects z similarly does not depend on the copper price. The prospects z added in the following examples are not constant. They correlate with the other prospects. In Table 1.5.3, we can obtain x þ z from supplying both ice cream and magazines under the assumption that the profits made from selling magazines are independent of the supply of ice cream, and vice versa. We can obtain y þ z from offering both hot dogs and magazines under similar independencies. Additivity is reasonable for moderate amounts of money. The extra receipt of z on your bank account does not change your situation much if z’s outcomes are moderate. It will, accordingly, not affect your preference between x and y. Only if the amounts

Chapter 1: Dutch books/No-arbitrage

24 Table 1.5.1

If then

x z xþz

no rain

some rain

all rain

400 500 900

100 500 600

400 500 100

≽ ≽

y z yþz

no rain

some rain

all rain

400 500 100

100 500 600

400 500 900

Notes: x: ice cream. y: hot dogs. z: message from the tax authorities that you receive a tax credit of $500. Table 1.5.2

If then

x z xþz

E1

E2

E3

50K 30K 80K

30K 30K 0

30K 30K 0

≽ ≽

y z yþz

E1

E2

E3

30K 30K 0

30K 30K 0

50K 30K 80K

Note: z: message from the tax authorities that you receive a tax credit of 30K. Table 1.5.3

If then

x z xþz

no rain

some rain

all rain

400 150 550

100 100 200

400 50 350

≽ ≽

y z yþz

no rain

some rain

all rain

400 150 250

100 100 200

400 50 450

Notes: x: ice cream. y: hot dogs. z: magazines.

in z are so large that they significantly affect your life and, thus, significantly affect the value of extra money for you, can the receipt of z matter for the choice between x and y.5 The following example illustrates this point. Example 1.5.2. Look at the finance Example 1.1.2 where you consider x and y. Now assume, however, that you wrote options before that together yield the benefits described by z in Table 1.5.4. Here z provides leverage for the payments of x and a hedge for those of y. Without z available you would prefer x to y. Assume that losses not exceeding 40K can be handled with no problem, but larger losses lead to liquidity problems. A loss of 70K even leads to bankruptcy. Then x þ z in Table 1.5.4 is too risky, implying x þ z ≺ y þ z. This preference together with x ≽ y fails additivity. The loss of 40K at event 5

If you know the concept of utility: It only happens if outcomes are so large that utility significantly deviates from linearity. This typically concerns outcomes exceeding a two-months salary. In a later chapter on expected utility we will show that a Dutch book and arbitrage are possible as soon as there is risk aversion (Assignment 3.3.6).

1.5 de Finetti’s Dutch books

25

Table 1.5.4

If

x z xþz

then?

E1

E2

E3

50K 40K 90K

30K 0 30K

30K 40K 70K

≽

y z yþz

≽?

E1

E2

E3

30K 40K 10K

30K 0 30K

50K 40K 10K

E3 generated by z has changed your situation and the value of money. You cannot afford losses from there on. □ The example demonstrated that large amounts of money at stake can affect your way of living and, hence, additivity need not be reasonable then (Arrow 1951a p. 431). It would be interesting to restrict the following analysis to a restricted domain of moderate amounts, where additivity is plausible. In the practice of finance, for instance, upper bounds are placed on the volume of assets that can be traded, and arbitrage opportunities can freely be searched only for limited financial transactions. For simplicity of analysis, we will examine the condition with the whole real line Ñ as outcome set in what follows. Note how we use superscripts rather than subscripts to index prospects because subscripts refer to events. Exercise 1.5.1.!b Assume that ≽ is transitive and additive. Prove: (a) [Improving one prospect in a sum of prospects improves the whole sum]. If xi ≽ yi then y1 þ    þ yi1 þ xi þ yiþ1 þ    þ yn ≽ y1 þ    þ yi1 þ yi þ yiþ1 þ    þ yn. (b) [Improving several prospects in a sum of prospects improves the whole sum]. If xi ≽ yi for all i then x1 þ    þ xm ≽ y1 þ    þ ym. Table 1.5.5 illustrates this result for m ¼ 2. Table 1.5.5 no rain some rain 1

If x 400 and x2 200 then x1 þ x2 600

100 100 200

all rain 400 0 400

≽ ≽ ≽

1

y y2 y1 þ y2

no rain

some rain

all rain

400 50 350

100 100 200

400 150 550

Notes: x1: ice cream. y1: hot dogs. x2: newspapers. y2: umbrellas.

□ We next introduce Dutch books, also called arbitrage. Using Exercise 1.5.1, it can be shown that, if a Dutch book exists, then the conditions just introduced (transitivity, additivity, monotonicity) cannot all hold true. You will be asked to show this in Exercise 1.5.2. Dutch books thus entail a particular way to violate the conditions introduced before.

26

Chapter 1: Dutch books/No-arbitrage (x11 , ... , x1n )

m (xm 1 , ... , x n )

+

∑xj1

+

… ∑xjn

. . .

(y11 , ... , y1n )

m (ym 1 , ... , y n )

+

∑yj1

+

… ∑yjn

Figure 1.5.1 Arbitrage (a Dutch book).

Definition 1.5.3. Arbitrage, or a Dutch book, consists of preferences xj ≽ yj, j ¼ 1, . . ., m such that the preferred prospects xj, when combined, (for P each state s) Palways m m j j j yield less than the nonpreferred prospects y . That is, j¼1 x ðsÞ < j¼1 y ðsÞ for all s ∊ S. (See Figure 1.5.1.) □ The x’s are individually judged preferable but jointly they turn out to be impoverishing under all states. We say that arbitrage is possible if arbitrage – or a Dutch book – exists; that is, there exist xj’s and yj’s as in Definition 1.5.3. Appendix 1.10 discusses mathematical differences between Definition 1.5.3 and the versions commonly presented in the literature. In particular, our definition does not involve scalar multiplication, so that a weaker preference condition and more general theorems will result. By Exercise 1.2.4b (partition flexibility), we may assume that all prospects refer to the same partition (E1,. . .,En), which we suppress. In financial markets, arbitrage cannot exist for an extended period. Assume it would, as in Figure 1.5.1. Then you obtain a sure and effortless profit by buying all y’s and selling all x’s. To see this point, first note that each xj is more expensive for the trades made. In than yj, so that more is paid to you  than you pay  to others l addition, you receive an amount yli þ    þ ym þ    þ xm  x for the event Ei i i i that will be true, and this amount is positive for all i. Thus the trading of prices to pay and receive has brought you a positive gain, and the portfolio you bought, being a Dutch book, also brings you a positive gain (no matter which state is true). You have indeed extracted a sure profit from the market while rendering no service in return, and you will continue repeating this transaction as long as the market allows it. Markets in such positions should quickly change or they will break down. For individual choice, a Dutch book in your preference system suggests an irrationality on your part: You would prefer to replace yj by xj for all j, but all together these replacements simply make you worse off no matter which state is the true one. Dutch books can be interpreted as violations of a conservation-of-preference principle: The preferences suggest that there is more total goodness comprised in the x-outcomes to the left, but the column-inequalities suggest the opposite, with more total goodness comprised in the y-outcomes to the right.

1.6 Behavioral foundation of EV

27

A Dutch book can be justified if there are interactions between outcomes. I explain the point briefly for interested readers in the rest of this paragraph. First, there may be interactions between outcomes “in the horizontal direction” when aggregating the outcomes of a prospect into an overall preference value of the prospect. For example, with n ¼ 3, an outcome M ¼ $1 million (instead of 0) contingent on event E3 may add more preference value when the outcomes under events E1 and E2 are also both M, such as in (E1:M,E2:M,E3:M) versus (E1:M,E2:M,E3:0), than when they are $5 million and 0, such as in (E1:5M,E2:0,E3:M) versus (E1:5M,E2:0,E3:0). In the former case the outcome is exactly what is required to bring certainty.6 Second, there may be interaction between outcomes in the vertical direction when adding them. Possibly an extra outcome M adds more goodness when added to an outcome $0 than when added to an outcome $4M.7 The following exercises concern the logical relations between Dutch books and preference conditions introduced before. Exercise 1.5.2.b Show that transitivity, monotonicity, and additivity preclude the existence of a Dutch book. □ Exercise 1.5.3. Suppose that ≽ is a weak order, that for each prospect a certainty equivalent exists, and that no Dutch book can be made. Derive the following claims (a and b together entail monotonicity). (a)a If x(s) > y(s) for all states s, then x  y; (b)c If x(s)  y(s) for all states s, then x ≽ y; (c)c Additivity holds. □ 1.6

A behavioral foundation of subjective probabilities and expected value using no-arbitrage and no-book

If you consider additivity to be an appropriate rationality requirement for your decisions then, given that transitivity and monotonicity are natural too and given the result of Exercise 1.5.2, you will not want to have a Dutch book in your preference system. It leads to the following theorem. Equivalence of statements in the theorem means that one statement holds if and only if the other does. That is, it refers to logical equivalence. Throughout this book, proofs are in appendices at the end of chapters. The following theorem is thus proved in Appendix 1.11. Theorem 1.6.1 [De Finetti; no-arbitrage]. Under Structural Assumption 1.2.1 (decision under uncertainty), the following three statements are equivalent. (i) Expected value holds. (ii) The preference relation ≽ is a weak order, for each prospect there exists a certainty equivalent, and no arbitrage (Dutch book) is possible. 6 7

In the terminology of }4.8, the sure-thing principle, designating separability between disjoint events, will be violated then. In the terminology of Chapter 2, the utility of money then will be nonlinear. Exercise 3.3.2 will elaborate on this point.

28

Chapter 1: Dutch books/No-arbitrage

(iii) The preference relation ≽ is a weak order, for each prospect there exists a certainty equivalent, and additivity and monotonicity are satisfied. □ Statement (i) claims existence of subjective probabilities such that expectation with respect to these can be taken to accommodate preference. At this stage, it may be conjectured that many different probability measures could serve this purpose. This is not so and the probabilities are unique, as stated next. Observation 1.6.10 [Uniqueness result for Theorem 1.6.1]. In (i) of Theorem 1.6.1, the probabilities are uniquely determined. □ Under statement (i), certainty equivalents and expected values are the same. In finance, Theorem 1.6.1 implies that market trades are as-if risk neutral. For individual choice theory it implies that, if the preference conditions in (ii) and (iii) are considered desirable, then uncertainties have to be expressed in terms of subjective probabilities. A detailed discussion of behavioral foundations such as Theorem 1.6.1 is in }1.8. The rest of this section presents examples and exercises. Example 1.6.2. Consider Table 1.6.1. The three events “no rain,” “some rain,” and “all rain” refer to tomorrow’s weather. Assume that a person uses the following decision strategy: when choosing between two prospects, prefer the prospect that yields a positive gain for most of the three events. It turns out that this person cannot be described by expected value maximization because a Dutch book can be made against him. Table 1.6.1 shows how a Dutch book can be made. Table 1.6.1 no rain some rain all rain 1

x x2 x3

0 100 100

100 0 100

100 100 0

x1 þ x2 þ x3 200

200

200

≽ ≽ ≽

no rain

some rain

all rain

y y2 y3

300 0 0

0 300 0

0 0 300

y1 þ y2 þ y3

300

300

300

1

□ In the vendor Example 1.1.1 and its subsequent discussions we modeled the three weather conditions as states of nature. It is usually not necessary to specify the states of nature. Thus, in Example 1.6.2, we need only specify that the three weather conditions concern events. We demonstrated that the combination of y’s yielded more for each of the three events considered. It then automatically follows that the combination of y’s yields more for every state of nature even if we did not specify those states of nature. The state space S will usually play such a hidden role, implicit and in the background, in this book. We will mostly specify events without specifying states of nature in this book. Exercise 1.6.1.b Consider the vendor Example 1.1.1. Assume that a person uses a maxmin decision strategy, valuing prospects by their worst outcome. Construct a Dutch book against this person. Here and later, you can use any prospect you like, and are not confined to those in Table 1.1.1. □

1.6 Behavioral foundation of EV

29

Exercise 1.6.2.b Consider Example 1.1.1. Assume that a person uses a maximax decision strategy, valuing prospects by their best outcome (“I choose the prospect for which I can gain most”). Construct a Dutch book against this person. □ Ad hoc threshold criteria such as those in the following exercise are popular in practical policy decisions and, for instance, in grading in teaching. They entail noncompensatory heuristics; i.e., heuristics avoiding weighting pros against cons, making them simple to implement but often leading to irrational decisions. Empirical studies into such heuristics include Brandsta¨tter, Gigerenzer, & Hertwig (2006), Birnbaum (2008b), and Glo¨ckner & Betsch (2008). Swait (2001) provides a model intermediate between compensatory and noncompensatory evaluations. Exercise 1.6.3b [Ad hoc threshold criteria]. Consider Example 1.1.1. Assume that a person maximizes EV with respect to the probabilities P(s1) ¼ P(s2) ¼ P(s3) ¼ ⅓, but adds an extra safety requirement: at most one of the possible outcomes may be negative. In other words, EV is modified in the following sense: if prospect x has two or more negative outcomes and prospect y has at most one, then y is preferred to x regardless of their EVs. For example, 100s1(1) ≺ (1)s11. Construct a Dutch book. □ Exercise 1.6.4.b Show that Statement (i) in Theorem 1.6.1 implies Statements (iii) and (ii) (deriving (iii) is easiest). □ The following exercise presents a highly appealing method for measuring subjective probabilities, using what are known as proper scoring rules. An explanation is given in the elaboration of the exercise. Exercise 1.6.5c [Proper scoring rules]. (a) Assume that you satisfy the conditions of Theorem 1.6.1, and that P denotes your probabilities. You have to choose a number r between 0 and 1 now, called your reported probability of E (rain tomorrow). Tomorrow you will receive the prospect (1(1r)2)E(1r2). Which number r do you choose (to be expressed in terms of P(E))? (b) Same question, only now you receive the prospect ln(r)Eln(1r) (so that you will lose money). □ The following exercise considers deviations from EV (EV is “risk neutrality”), with risk aversion specified in part (a) and risk seeking specified in part (b). Any such deviation directly implies arbitrage. Exercise 1.6.6.b Assume that a coin will be tossed, giving heads (H) or tails (T). Assume weak ordering, with aHb  bHa for all outcomes a,b. This symmetry condition can be interpreted as H and T having probability ½. (a) Assume risk aversion in the sense that there exist g > b such that CE ¼ CE(gHb) < (bþg)/2. Show that arbitrage is possible. (b) Assume risk seeking in the sense that there exist g > b such that CE ¼ CE(gHb) > (bþg)/2. Show that arbitrage is possible. □

30

Chapter 1: Dutch books/No-arbitrage

Because the result of the following exercise is conceptually important, the elaboration is immediately given in the main text. It shows that, if nonarbitrage and the other conditions in Theorem 1.6.1 are necessary conditions for rational choice, they surely are not sufficient. Exercise 1.6.7.!a Assume a symmetric six-sided die with Ej the event of j coming up. Paul and everybody agree that each Ej has an objective probability of 1/6 of coming up. Paul maximizes EV as in Theorem 1.6.1, but his subjective probabilities, denoted Q here, differ from the objective ones denoted P. He has Q(E1) ¼ ½, Q(E2) ¼    ¼ Q(E6) ¼ 1/10 and is willing to take any bet favorable according to his judgment, such as 100E1(99). Can you make a book against Paul? Elaboration. Paul maximizes EV with respect to Q so that, by Theorem 1.6.1, no book can be made against Paul. Nevertheless, his behavior does not seem to be wise. Behavior that is not vulnerable to arbitrage can still be silly by, for instance, using silly probabilities. In the literature, overly strict interpretations of what is called the coherence viewpoint have sometimes been advanced (de Finetti 1931b; properly discussed by Tversky & Kahneman 1981 p. 458; for recent references, see Olsson 2005, and Hammond 2006). These interpretations claim that preferential consistency conditions are not only necessary, but also sufficient, for rational behavior (see Appendix B.1). □ The following assignment presents a generalized way of measuring subjective probabilities, and gives a behavioral foundation to EV through the requirement that these measurements do not run into contradictions. Such behavioral foundations, using requirements of consistency of measurement of subjective parameters, are central to the rest of this book. For the following assignment, as for all exercises unless explicitly stated otherwise, you can use all theorems, exercises, and results that preceded it. In particular, you can use the equivalences established in Theorem 1.6.1. Assignment 1.6.8.a Show that the following statement is equivalent to each statement in Theorem 1.6.1: Monotonicity holds, for each prospect a CE exists that represents preference, and the CE function is additive (CE(x þ y) ¼ CE(x) þ CE(y)). □ Assignment 1.6.9.c As demonstrated in Eqs. (1.3.1) and (1.3.2), an equivalence p  1E1 0

ð1:6:1Þ

implies P(E1) ¼ p. A drawback of this particular measurement is that a riskless amount is compared to an uncertain prospect, and it is well known that such choices are prone to many biases. Hence an alternative measurement, through E1 ðx1 þ aÞ; E2 x2 ; . . . ; En xn  E1 ðx1 þ paÞ; . . . ; En ðxn þ paÞ; ða 6¼ 0Þ

ð1:6:2Þ

and also implying P(E1) ¼ p under EV, can be useful. For EV to hold, measurements of Eqs. (1.6.1) and (1.6.2) should not lead to contradictions and should

1.6 Behavioral foundation of EV

31

give the same p value. Hence, a necessary preference condition for EV is the following implication: ½Eq: ð1:6:1Þ ) Eq: ð1:6:2Þ for all E1 ; . . . ; En ; x1 ; . . . ; xn ; and a 6¼ 0:

ð1:6:3Þ

Equation (1.6.2) enables us to avoid riskless outcomes when measuring subjective probabilities. Equation (1.6.3) is a weakened version of additivity, restricting the implication x ≽ y ) x þ z ≽ y þ z to indifference () and imposing it only for x and y as in Eqs. (1.6.1) and (1.6.2). Assume weak ordering, monotonicity, and the existence of a CE for every prospect. Show that Eq. (1.6.3) holds if and only if EV holds. If you do not see how to do this, you can find a hint in the footnote.8 □ Assignment 1.6.10a [Tversky & Kahneman 1981 p. 454, with underlying events added]. Consider an event E with known probability ¼. Its complement has probability ¾. In an experiment of Tversky & Kahneman (1981), a majority of subjects expressed the following preferences: 240E 240 ≽1000E 0 0E 1000 ≽ 750E 750: Show that these preferences entail arbitrage. Tversky & Kahneman (1981) offered both choices simultaneously to their subjects, so that the arbitrage effectively took place and a straight violation of monotonicity resulted. It makes it extra remarkable that these majority preferences were still found. □ Assignment 1.6.11b [The put-call parity in finance]. In deviation from the conventions in the rest of this book, we consider prospects with infinitely many outcomes in this assignment. Similar to Example 1.1.2 (finance), we assume that the state space S is [0,1), describing the copper price per pound next month. We consider three prospects, x(s) ¼ s  10K 25K, y(s) ¼ max{x(s), 0}, and z(s) ¼ min{0, x(s)}, depicted in the figure. We assume zero discounting.

Payoff next month

2.50

8

Payoff next month

x(s)

s

Payoff next month

y(s)

2.50

s

z(s)

2.50

s

It can be seen that in the proof of Theorem 1.6.1 we never need more than this weak version of additivity. Figure 1.11.1 only needs x and y as in Eqs. (1.6.1) and (1.6.2) and Lemmas 1.11.1 and 1.11.2 can also be restricted to such cases, where further only indifferences are involved. Thus, additivity in Statement (iii) of Theorem 1.6.1 can be weakened to Eq. (1.6.3).

Chapter 1: Dutch books/No-arbitrage

32

In finance, x(s) designates a forward contract committing you to buy 10,000 pounds of copper at (“strike”) price $2.50 a pound next month; i.e., it requires a payment of 25K for something worth s  10K. The payoff next month is, therefore, (s  10K)  25K. y(s) designates a (European) call option where you have the right, at your discretion, to buy 10,000 pounds of copper for a (strike) price of $2.50 a pound next month. Its payoff next month will be 0 if the market copper price s will be below $2.50, and it will be s$2.50 per pound if s will exceed $2.50. z(s), finally, designates a short position in a (European) put option. This means that another agent has the right, at his discretion, to sell 10,000 pounds of copper for a (strike) price of $2.50 a pound to you next month, and will do so only if the market prices will be below $2.50. Assume that the market prices of x, y, and z today are X, Y, and Z, respectively. Show that no-arbitrage implies the following equality, not by using de Finetti’s Theorem 1.6.1, but by directly constructing arbitrage from violations of the equality. X ¼ Y þ Z ½The put-call parity

ð1:6:4Þ

□ Assignment 1.6.12a [Race track betting]. In the 1986 Kentucky derby, a horse (“Ferdinand”) paid $8.40 per $1 at a bookmaker (“Hollywood Park”) in California, and $18.70 at a bookmaker (“Aqueduct”) in New York (Thaler & Ziemba 1988 p. 167). Let E be the event of the horse winning. Assume that you can buy or sell as many gambles on the horse as you like. That is, you can receive or give away, at your discretion, the prospects 7.40E1(1.00) and 17.70E1(1.00) as well as multiples of these. Demonstrate that there is a possibility for arbitrage. □ The most important condition in de Finetti’s bookmaking argument is the additivity condition. Yet de Finetti mostly stressed monotonicity. The following assignment shows how violations of additivity can be presented as violations of monotonicity. Assignment 1.6.13c [Turning violations of additivity into a book]. Assume that there exists an m and x1,. . .,xm, y1,. . .,ym such that x1 ≽ y1,. . .,xm ≽ ym, and x1 þ . . . þ xm ≺ y1 þ . . . þ ym. By Exercise 1.5.1b, this entails a violation of additivity. It, in itself, is no book however because the sum of yi’s need not dominate the sum of xi’s in every state. Show that a book can nevertheless be made. You may assume the following continuity in this assignment: If x  y, then for some small positive ∊, x  y þ ∊. □ 1.7

Discussion of Structural Assumption 1.2.1 (Decision under uncertainty)

Throughout, we require that the decision maker does not have any influence on the truth of the events (Dre`ze 1987 }1.2.1; Machina 1987 p. 148). Gilboa (2009 }2.6) wrote: “The distinction between the acts, over which the decision maker has control, and states, over which she hasn’t, is one of the pillars of rational choice. Many mistakes in everyday life have to do with an inappropriate distinction between acts and states.” In the elections Example 1.1.3, you have no right to vote. In the finance

1.7 Structural Assumption discussed

33

Example 1.1.2, you are a price taker (economic term for a consumer who cannot influence prices). In the vendor Example 1.1.1, you are not a rainmaker who can generate rain tomorrow and can, thus, make state of nature s3 come true. The absence of influence on the truth of events implies, in particular, that there is no moral hazard. Moral hazard originated from insurance, where it designates an increase in the risk of loss or damage caused by becoming more careless when insured. It is nowadays used as a general term to refer to situations where the likelihood of uncertain events can be influenced. The techniques of this book cannot be applied directly to such cases. However, they can be applied after a proper remodeling of the state space, incorporating all relevant influences you have in the prospects to be chosen and specifying as states and events the uncertainties completely outside your control. Such remodelings are useful in making essential aspects of the situation explicit. In many practical situations, obtaining a strict separation between what is completely under your control (which prospect is chosen from those available) and what is completely outside of your control (states, events) is nontrivial. If you are a student taking an exam, then you have partial influence on the grade you will get, and states of nature and events cannot directly refer to your future grade. It is then not easy to specify what exactly your own influence is and what the uncertainties are outside your influence. Your grade does become an event outside of your control, and amenable to probability theory, if you condition on your influence and, for instance, assume that you will make the maximal effort, or a regular effort. The strict “Cartesian” separation of what is under your influence and what is outside your influence is the key in Savage’s (1954) model, as it is in the many studies, including this book, that build on Savage’s model. To understand the limitations of this model, it is instructive to study alternative models, such as Jeffrey’s (1965) thought-provoking one, or Lewis’ (1987) caused decision theory. In the brilliant early contribution by Ramsey (1931), no separation as just described was made in any clear manner. (Arrow 1951a, p. 424 described Ramsey’s contribution as “none too clear.”) Discussions of the limitations include Ahn (2008), Dre`ze (1987 }1.2), Fishburn (1970 p. 161, }12.1; p. 166, }12.2; p. 168, end of }12.2), Gilboa (2009 Ch. 11), Jeffrey (1965), Karni (2006), Krantz et al. (1971 }8.2.1), Kreps (1988 pp. 35–38), Luce (2000 }1.1.6.1), and Nau (2001). In our structural assumption, states and outcomes are taken as primitives and prospects are constructed from these, mapping states to outcomes. Consequently, the prospect set is larger than the state space or the outcome space. In applications, some prospects, states – or events – and outcomes will be naturally given, and others are defined in terms of these. We may, for instance, sometimes take a state and a prospect and define as outcome whatever results if that state obtains and that prospect is chosen. Or we can easily specify the conceivable prospects and the conceivable outcomes, but not the states of nature. Then we may define a state by specifying for each prospect which outcome will result (Gilboa 2009 }11.1.2). Savage’s assumption that consequences are portable across states need not hold in the latter approaches.

34

Chapter 1: Dutch books/No-arbitrage

In all aforementioned constructions, the resulting model will contain some artificially constructed objects that do not match conceivable real objects. There is no clear recipe for the best way to model an actual decision situation, and modeling is an art that has to be learned in practice. Works besides the aforementioned references that have challenged Savage’s way of modeling include Green & Osband (1991) and Krantz et al. (1971 }9.6.3). Exercise 1.7.1.b Suppose that you have to decide whether or not to smoke for the rest of your life. You construct the model in Table 1.7.1. Table 1.7.1

smoke don’t smoke

lung cancer

no lung cancer

pleasant life, then disease unpleasant life, then disease

pleasant life, healthy unpleasant life, healthy

Outcomes are assumed monetary in the main text with a few exceptions that are noted explicitly. This example concerns such an exception. You decide from this model that smoking gives a superior outcome for every state of nature, so that by monotonicity you would decide that smoking is to be preferred to not smoking. Obviously, something is wrong with this model. Can you point out what? □ 1.8

The general nature and usefulness of behavioral foundations

This section elaborates on the discussion of behavioral foundations in the introduction to this book. Related discussions are in Gilboa, Postlewaite, & Schmeidler (2008 pp. 177–178) and Gilboa (2009 }8.4). Behavioral foundations are mathematical results, stating a logical equivalence between a list of preference conditions and a decision model. They show that the subjective parameters in the decision model (subjective probabilities in Theorem 1.6.1) are the relevant parameters for determining decisions. They change the status of such subjective parameters from being ad hoc into being scientifically well founded. Historical examples. Already in the 1930s, there had been many discussions of the plausibility and implausibility of the concept of subjective probabilities, often based on philosophical arguments. This topic was central in the debates on Bayesian versus frequentist statistics. De Finetti (1931a), Ramsey (1931), and Savage bypassed such philosophical debates and instead showed that subjective probabilities must exist by giving them behavioral foundations. For models that did not receive preference foundations, this absence is usually considered a drawback, and the unfortunate history of the separate-probabilitytransformation model serves as a warning (details in Chapter 5). The well-known Analytical Hierarchy Process, Saaty’s (1980) popular and pragmatic technique for multiattribute decisions, has often been criticized for such an absence

1.8 Behavioral foundations: Their use

35

(Smith & von Winterfeldt 2004). Birnbaum’s (2008a) models that fitted data better than prospect theory in several experiments, also lack a preference foundation and have not been widely used in theoretical works. The same holds for Busemeyer’s decision field theory (Busemeyer & Diederich 2002). Although the multiple priors model had been available since Wald (1950) and Hurwicz (1951), it only became popular when Gilboa & Schmeidler (1989) provided a preference foundation for the case of linear utility. Convincing people of the prescriptive appropriateness of a model. When teaching, I use the following thought experiment to explain the usefulness of behavioral foundations. Imagine that someone, a decision analyst, out of the blue advises you to make your decisions under uncertainty in agreement with EV. That is, you should choose probabilities (the Ps in the EV maximization of Theorem 1.6.1, which can be subjective), then take the probability-weighted average of the outcomes as evaluation of each prospect, and then, finally, choose from all available prospects the one with the highest value calculated in this manner. Such is the advice. As it stands, the advice would be ad hoc and unconvincing. Why would you express uncertainties in terms of probabilities, and why would you carry out the calculations as recommended? Why not take squares or exponents of probabilities, and why sum the products of probabilities and outcomes? An additional problem is that it is not very clear from the advice of the decision analyst how the probabilities should be chosen. The probabilities seem to come like manna from heaven. The advice, if presented in the ad hoc manner as just described, would be unconvincing. Imagine now that the decision analyst apologizes for having given you such unconvincing advice, and then changes the topic of conversation. He recommends that your preferences satisfy completeness and transitivity, and you, being a rational person, agree. You also agree that the existence of a certainty equivalent for each prospect is reasonable. Then, the decision analyst explains to you what a Dutch book is. You do not want your preference system to contain a Dutch book. Then, however, a surprise will be awaiting you. The decision analyst says, triumphantly, that he has convinced you of his first claim (EV maximization) after all, and comes up with Theorem 1.6.1. This theorem proves that, by accepting some seemingly natural conditions on your preferences, your behavior can be described, like it or not, as in Statement (i). In particular, there are probabilities representing your uncertainties. You behave in accordance with Statement (i), consciously or subconsciously. This thought experiment shows how behavioral foundations can convince people to use particular quantitative models of decision making. The models in themselves are not directly meaningful9 in an empirical sense. It is not clear why Statement (i) of Theorem 1.6.1 in isolation without further justification would lead to wise behavior. Statement (ii) shows why it can do so. Statement (ii) is stated directly in terms of preference conditions, which makes it directly empirically meaningful. In finance,

9

Formalizations of empirical meaningfulness are in Narens (2002) and Pfanzagl (1968 Ch. 2).

36

Chapter 1: Dutch books/No-arbitrage

“risk neutral” evaluations as in Statement (i) are similarly justified by the absence of arbitrage as in Statement (ii). Criticizing a model. In general, a behavioral foundation gives necessary and sufficient behavioral conditions – conditions stated directly in terms of your preferences – for a general kind of quantitative model. In the preceding discussion, the behavioral foundation was used to convince a person that a model is appropriate by arguing that the preference conditions are appropriate. Behavioral foundations can equally well be used to criticize models. Maybe you felt that the method of Statement (i) appeared to be sensible at first sight, but then you decided that the existence of Dutch books is sometimes appropriate still. Theorem 1.6.1 would then convince you that the EV model of statement (i) there, with probabilities, is not appropriate after all. Similarly, Allais’ (1953a) famous criticism of expected utility (presented in }4.12) was targeted on the independence condition, the most critical condition in behavioral foundations of expected utility. Descriptive use in the experimental heaven. In teaching, I refer to the “experimental heaven” for explaining the descriptive use of behavioral foundations. There are unbounded measurement facilities in the sense that we can observe all, infinitely many, preferences, and we can analyze them all. Further, there are no measurement errors so that all preferences are observed exactly and we can use a deterministic model. A descriptive model can then readily be verified or falsified using the conditions in a behavioral foundation. In reality, of course, only a limited number of observations can be made, and we must reckon with measurement errors so that we have to use statistical tools. Identifying relevant parameters. Example 3.7.1 will illustrate another use of behavioral foundations, through an example with chronic health states (Q,T), designating T years of life in health state Q followed by immediate death. It may not be very clear, a priori, what kind of model should be used to evaluate chronic health states, and which concepts are relevant. A behavioral foundation (Observation 3.7.2) will show how plausible preference conditions imply that we should use a multiplicative model defined by U(Q,T) ¼ V(Q)W(T). Here V measures quality of life and W measures the value of life years (incorporating discounting). This model is the famous QALY model, used throughout the health domain. In this way, behavioral foundations help to identify the relevant parameters and to measure value in situations where it is not clear, a priori, how to do so. Gentlemanly distant? The aforementioned reasons for developing behavioral foundations do not specify whether the preference conditions in a behavioral foundation should be close in nature to the corresponding preference functional, or whether it is preferable that they are, or at first seem to be, of a different nature. In mathematics, it is usually considered desirable that conditions that are logically equivalent to some hypothesis are “gentlemanly distant” from this hypothesis, because otherwise the logical equivalence is too trivial and unsurprising and does not give many new insights. For these reasons it may be preferred that preference conditions do not resemble the corresponding functional very closely.

1.9 Kinds of preference conditions

37

There are reasons that make a close resemblance between preference conditions and the corresponding functional desirable after all. The preference conditions then better clarify the empirical meaning of the preference functional, and better show the empirical meaning of parameters in the functional. Indeed, the technique for obtaining behavioral foundations used in this book, deriving them from the exclusion of contradictions in empirical measurements, will mostly lead to preference conditions that are close to the corresponding functionals. They thus help to illustrate that the functionals are homeomorphic to the decision process (see Introduction). For example, the additivity preference condition in Theorem 1.6.1 is close to the additive probabilities and expectations used in the representing functional, and in that sense the theorem may not be very surprising. I think that, in return, this additivity preference condition better captures the empirical meaning of subjective probabilities than any other condition that I can think of, implying that choices can be isolated from many aspects of the context. The purpose of preference foundations is to identify empirical content, and not to be mathematically complex. The simpler the better. I hope that the mathematically sophisticated readers are willing to accept, in return, the drawback (from their perspective) that the mathematics in this book is simpler than in most other studies on behavioral foundations. De Finetti’s electrifying writing. When I, as a mathematics student in 1978, expressed amazement about the claim made by my statistics teacher – a frequentist as I now know – that the probability of life on Mars could not be defined, and that it was even treated differently than the probability of a random repeatable event such as related to coin tosses, he referred me to Bruno de Finetti’s work. (“There is a crazy Italian who thinks such things” were his exact words.) De Finetti’s (1972) first chapter, written in a thought-provoking manner, opened up to me the technique of behavioral foundations, and the possibility of defining something as seemingly intangible as one’s subjective degree of belief, in a tangible manner. De Finetti showed how we can read the minds (beliefs, i.e. subjective probabilities) of people. I felt electrified by his ideas, and decided that I wanted to work on them. I hope that the readers will also sense some of the magic of these ideas.

Appendix 1.9

Natural, technical, and intuitive preference conditions, and the problematic completeness condition

Natural and technical preference conditions. Some natural preference conditions – transitivity and monotonicity – are assumed throughout this book. In addition to those natural preference conditions, the behavioral foundations in this book adopt some technical richness assumptions, entailing that the set of prospects constitutes a continuum. By adding continuity assumptions, we can then carry out measurements to any desired degree of precision. Continuity was incorporated in this chapter by the assumption that a certainty equivalent exists for each prospect. We have assumed throughout that the outcome set is a continuum. In Chapter 2 this assumption could be

38

Chapter 1: Dutch books/No-arbitrage

dropped – although we will maintain it – because we will have a continuum of probabilities there. Three problems for completeness. Throughout, we have assumed completeness of preference. One reason that has often been advanced in the literature for violating completeness concerns indeterminacy of preference: agents cannot make up their minds in a decision situation they face and are undecided which of two prospects to choose (Bewley 1986; Eliaz & Ok 2006; Gilboa, Postlewaite, & Schmeidler 2008 pp. 179–180; Levi 1980; Mandler 2005; Walley 1991). We will assume throughout that an agent simply has to choose one prospect in every decision situation faced (Gilboa 2009 }6.1). A prospect “doing nothing,” if available, is simply a choice option available in the set to be chosen from. In this sense, our completeness assumption can be taken as one more natural preference condition of the same kind as transitivity and monotonicity. A second argument against completeness concerns the unrealistic nature of some prospects in our models. We assume that all assignments of outcomes to events are available in the preference domain. However, some of these assignments may be too artificial to even be conceivable. In a famous discussion, Savage gave the example of “being hung so long as it be done without damage to my health or reputation” (Dre`ze 1987 p. 78, Appendix 2.A; discussed by Gilboa 2009, }12.2.2). Assuming preferences over such prospects is indeed problematic. Even if two prospects are both conceivable, a choice situation where both are available may still be inconceivable. For example, a choice between fries with curry sauce to be served and consumed immediately, or foie gras to be served and consumed immediately, may not be realistic if these dishes are only served at places far apart. A third argument against completeness concerns the richness of the models assumed, that constitute continuums, with choices between all prospect pairs assumed observable. We will never observe infinitely many data, let alone continuums (Davidson & Suppes 1956). Here completeness is an idealization that we make to facilitate our analyses. Although it has sometimes been suggested that completeness and continuity for a continuum-domain are innocuous assumptions (Arrow 1971 p. 48; Dre`ze 1987 p. 12), several authors have pointed out that these assumptions do add empirical (behavioral) implications to other assumptions. It is, unfortunately, usually unclear what exactly those added implications are (Ghirardato & Marinacci 2001b; Krantz et al. 1971 }9.1; Pfanzagl 1968 }6.6; Schmeidler 1971; Suppes 1974 }2; Wakker 1988).10 Behavioral conditions. Besides the natural and technical assumptions made throughout, behavioral foundations use preference conditions typical of the model considered, such as the bookmaking argument or additivity in this chapter. Such a condition will, for example, be independence – or its modified version called SG 10

In Savage’s model, the second and third argument are aggravated because he assumed that the state space specify all uncertainties relevant to the decisions. This assumption has caused many misunderstandings, and we will not follow it. We will only consider what Savage (1954) informally called small worlds.

1.10 Literature

39

consistency – for the von Neumann–Morgenstern expected utility model studied in Ch. 2. Such conditions are referred to as the intuitive conditions, and they are the focus of attention in the analysis of a theory. The natural and behavioral conditions are usually necessary for the basic decision model that interests us in the sense that they are implied by that model. General discussion and necessity and sufficiency of conditions. We would hope to find natural and behavioral conditions that are necessary and sufficient for the decision model of interest, at least for finite models. Unfortunately this task is, usually, too difficult (so-called first-order logic axioms cannot do it: Suppes 1974 p. 164). We, therefore, usually add the technical richness conditions, which simplify our task. Whereas the natural and intuitive conditions in isolation are only necessary for the decision model of interest (this necessity is usually easy to prove), together with the technical conditions they become necessary and sufficient (which is usually hard to prove). I consider the second problem of completeness (the unrealistic nature of some prospects), aggravated by the richness of the model assumed through the technical conditions, to be the weakest point of the revealed-preference paradigm and of the theoretical results of this book. The third problem of completeness (we can only make finitely many observations) is also troublesome. These two problems make it desirable to develop results for incomplete preferences defined on finite domains. Obtaining such results is, in my opinion, the major challenge for the theory of behavioral foundations to date. It can, however, be extremely difficult (see Shapiro, 1979 for subjective expected utility), which is why this book focuses on conventional models with completeness on continuums of preference domains. For other discussions of the different kinds of preference conditions, see Suppes (1974 }2), Krantz et al. (1971 }1.4 and }9.1), and Pfanzagl (1968 }6.6). Appendix 1.10

The relation of Theorem 1.6.1 to the literature

In finance, de Finetti’s Theorem 1.6.1 implies that market trades can be analyzed as-if risk neutral. For individual choice, Theorem 1.6.1 has presented de Finetti’s book idea as a coherence condition. De Finetti formulated his idea differently. He considered a game context, where the decision maker (you) has to play a role similar to the market in finance. That is, you have to make all your preferences public, for example by making all your certainty equivalents public. An opponent can take you up on any of your preferences xj ≽ yj, j ¼ 1,. . .,n, and ask you to exchange yj for xj with him for each j, which means that you give him (y1x1) þ . . . þ (ynxn). Then your opponent, even if absolutely ignorant about the states of nature, can make a sure profit from you (arbitrage) whenever a Dutch book exists in your preferences, that is, whenever you do not go by subjective probabilities and expected value. De Finetti thus formulated the bookmaking argument as a game with an opponent where the mechanism has been devised such that out of self-interest it is best to reveal

40

Chapter 1: Dutch books/No-arbitrage

private information about subjective probabilities. De Finetti’s formulation is more vivid than ours. In his formulation you will surely lose money to a cunning opponent the moment your preference system contains a Dutch book. It is, however, less clean because it is distorted by dynamic and game-theoretic complications. These are most problematic if your opponent has information about the states of nature that you do not have. Then your opponent can make money from you whatever you do, and the bookmaking game is a lost game anyway. It then seems to be immaterial whether or not your preference system contains a Dutch book, and the book-thought-experiment is not very interesting. Finally, de Finetti used the terms fair price or prevision rather than our term certainty equivalent. In finance, it is often assumed that no person can be better informed than the market, or can only be so for a short time period (the efficient market hypothesis). The moment someone knows that a stock is underpriced, this person starts buying the stock and its price will move up until it has reached its proper level. In this way, the market automatically aggregates all information possessed by its players into the market prices, as if driven by an invisible hand. To articulate more clearly the individual coherence interpretation of Theorem 1.6.1, imagine that you are alone in a hotel room some evening. Tomorrow you can choose one prospect from a pair of available prospects, being the one you prefer most, but you do not know which pair of prospects it will be. You do know that the payments will be moderate. You know your entire preference system over all prospect pairs, having the unbounded calculation power to know all these infinitely many preferences. If you know that your preference system contains a Dutch book, should you consider this to be irrational and should you change it? De Finetti’s formulation does not apply here because there is no other person involved who could benefit from your preferences in some sense. Another reason why de Finetti’s thought experiment does not apply is that only one of your preferences will be realized, rather than several preferences as in de Finetti’s formulation.11 The latter reason also implies that no law of large numbers applies. The coherence principles can still convince you to change your preferences. If x  y, but after addition of z we have x þ z ≼ y þ z whereas you think that z is too moderate to change your preference between x and y, then this can convince you that at least one of the two preferences does not give what best serves your interests, and that these preferences should be reconsidered. You thus satisfy additivity and behave as-if maximizing EV using subjective probabilities despite the fact that you face a oneshot individual decision. In finance, and also in de Finetti’s work, not only addition, but also scalar multiplication is considered in the constructions of arbitrage. Trades l1(x1y1) þ    þ ln(xnyn) are allowed in the arbitrage Definition 1.5.3, where the lj’s are nonnegative 11

Appendix B will explain that repeated choice is not an explicit part of the revealed-preference paradigm, and }B.3 will discuss the (absence of) repetition for expected value maximization. This point is central to Cubitt & Sugden’s (2001) discussion of bookmaking arguments.

1.11 Proof

41

real numbers at the discretion of your opponent (or an agent in the market).12 Theorem 1.6.1 generalizes these results by showing that scalar multiplication need not be invoked, and that it is implied by the other conditions. This generalization was introduced by Diecidue & Wakker (2002). I find the mere summation of prospects and preferences more satisfactory for individual choice than scalar multiplications, because the former can directly be interpreted in terms of a joint receipt of prospects. De Finetti also showed how the requirement of completeness of preference over all prospects can be restricted, in mathematical terms, to the linear space spanned by your expressed preferences (de Finetti 1974 Ch. 3). In finance, attention is also often restricted to incomplete markets with only particular stocks and their riskless hedges. Then an as-if risk neutral evaluation is obtained only for these, and risk aversion for other portfolios is not excluded. We shall not elaborate on these points.

Appendix 1.11

Proof of Theorem 1.6.1 and Observation 1.6.10

Hopefully the proof presented here will be appealing to the readers. I invested much time into getting the most appealing proof of de Finetti’s theorem, in particular through Figure 1.11.1. It was de Finetti’s theorem that brought me into this field. The uniqueness in Observation 1.6.10 follows from the indifference 1E0  P(E), plus the uniqueness of P(E) in view of the implied monotonicity. From now on, we focus on Theorem 1.6.1. Exercise 1.5.2 has demonstrated that transitivity, monotonicity, and additivity preclude the existence of a Dutch book. That is, (iii) in the theorem implies (ii). By Exercise 1.5.3, the absence of Dutch books together with weak ordering and the existence of CEs implies additivity and monotonicity; i.e., (ii) in the theorem implies (iii). Hence, it suffices to derive equivalence of (i) and (iii). Exercise 1.6.4 demonstrated that (i) implies (iii) (and also (ii)). In the rest of the proof, we assume (iii) and derive (i). Figure 1.11.1 shows the main steps in the proof. By Exercise 1.2.5b, CE represents preference, which we will use many times. The following lemma considers combinations of prospects. Lemma 1.11.1 [Combining prospects]. CE is additive; i.e., CE(f þ g) ¼ CE(f) þ CE(g). Proof. f  CE(f) and g  CE(g) ) f þ g  CE(f) þ CE(g), by Exercise 1.5.1b with m ¼ 2, and applied both with ≽ for  and with ≼ for . Hence CE(f þ g), the constant amount equivalent to f þ g, must be CE(f) þ CE(g). □ An alternative way to prove Lemma 1.11.1, and lemmas to follow, is by setting up arbitrage against any violation. Additivity is also known as Cauchy’s functional 12

de Finetti mostly considered the special case where yj is the certainty equivalent of xj so that the preferences are indifferences and the lj’s can be any real number (de Finetti 1974). Further, he sometimes (e.g. de Finetti 1931a) restricted attention to the case where the xj’s are indicator functions of events so that the certainty equivalents yj reflect subjective probabilities. In the presence of other conditions such as monotonicity and the linear combinations considered, these restricted conditions do still imply EV and the conditions described in the text.

42

Chapter 1: Dutch books/No-arbitrage E1

Decomposing a general prospect

E2

En Amplifying event indicators

E1

x1 x2

=

E1

x1

E2

E2

0 +

E1

0 x2 +

.

.

.

+

E2

. .

. .

. .

. .

.

.

.

.

xn

En

En

0

En

0

∼ P(E2)x2

∼ P(E1)x1

0 0

xn

∼ P(En)xn ∼

Combining prospects

P(E1)x1

+

P(E2)x2

+

.

.

.

+

P(En)xn

Figure 1.11.1 Deriving expected value.

equation. It is well known that under weak additional assumptions (e.g. monotonicity suffices) additivity implies that CE must be linear (Acze´l 1966 Theorem 2.1.1.1). The remainder of this proof establishes this result in our context. We call prospects of the form 1E0 event indicators. We define P(E) ¼ CE(1E0) for all events E. That is, P(E) is the CE of the event indicator of E. P is a probability measure because • • • •

P(S) ¼ CE(S:1) ¼ 1; P() ¼ CE(S:0) ¼ 0; 0 P(E) (by monotonicity); For disjoint A,B, P(A [ B) ¼ CE(1A [ B0) ¼ (by additivity of CE; that is, by Lemma 1.11.1) CE(1A0) þ CE(1B0) ¼ P(A) þ P(B).

Lemma 1.11.2 [Amplifying event indicators]. CE(lE0) ¼ P(E)l for all E and l. proof. case 1 [l ¼ 0]. Follows immediately. case 2 [l ¼ k for a natural number k]. By repeated application of Lemma 1.11.1, CE (kE0) ¼ CE(1E0 þ    þ 1E0) ¼ CE(1E0) þ    þ CE(1E0) ¼ P(E)k for all natural numbers k. case 3 [l is positive and rational; i.e., l ¼ m/k for natural numbers k and m]. CEðm=kE 0Þ ¼ PðEÞm 1k because (by Case 2) P(E)m ¼ CE(mE0) ¼ CE(m/kE0 þ    þ m/kE0) (k terms) ¼ CE(m/kE0) þ    þ CE(m/kE0) ¼ kCE(m/kE0). case 4 [l is negative and rational]. Follows from the result for all positive rational l and the equality CE(f) ¼ CE(f) for all prospects f. The latter equality follows because 0 ¼ CE(f  f) ¼ CE(f) þ CE(f).

1.11 Proof

43

case 5 [Nonrational l]. The derivation for this case does not need continuity, and mainly uses monotonicity. P(E) can be zero or positive in what follows. By monotonicity, for all rational ‘ (“low”) and h (“high”), ‘E0 ≼ lE0 ≼ hE0 whenever ‘ < l < h. Hence, P(E)‘ ¼ CE(‘E0) CE(lE0) CE(hE0) ¼ P(E)h. P(E)l is the only real number CE(lE0) that satisfies these inequalities for all ‘ < l < h. □ Figure 1.11.1 demonstrates, for a general prospect E1x1. . .Enxn (depicted left in the figure), that its CE is its expected value P(E1)x1 þ    þ P(En)xn, so that the latter formula represents preference. This completes the proof of Theorem 1.6.1. We obtain the following corollary, a result that is usually added as an assumption in analyses of arbitrage in finance, as it was in de Finetti’s analyses. We have demonstrated that it follows from the other assumptions. Corollary 1.11.3 [Amplifying general prospects]. CE(lf) ¼ lCE(f). □

2

Expected utility with known probabilities – “risk” – and unknown utilities

The important financial decisions in our life concern large stakes, and then the maximization of expected value may not be reasonable. Most of our decisions concern nonquantitative outcomes such as health states. Then expected value cannot be used because it cannot even be defined. For these reasons, a more general theory is warranted. We now turn to such a theory – expected utility. For simplicity, we consider only the case where probabilities are known in this chapter. This case is called decision under risk. The general case of both unknown probabilities and unknown utility is more complex, and will be dealt with in later chapters. Whereas Chapter 1 showed how to read the minds (beliefs, i.e. subjective probabilities) of people, this chapter will show how to read their hearts (happiness, i.e. utility). 2.1

Decision under risk as a special case of decision under uncertainty

Probabilities can be (un)known to many degrees, all covered by the general term uncertainty. Decision under risk is the special, limiting case where probabilities are objectively given, known, and commonly agreed upon. Risk is often treated separately from uncertainty in the literature. It is more efficient, and conceptually more appropriate, to treat risk as a special case of uncertainty. I will discuss this point in some detail in this and the following sections. This point will be especially important in the study of ambiguity in Part III. Machina (2004) provided a formal model supporting this point. Throughout this chapter and in fact throughout this whole book, Structural Assumption 1.2.1 (decision under uncertainty) is maintained. Assumptions will be added. All results from uncertainty can immediately be applied to risk, and results from risk can often be extended to uncertainty. I commonly advise students to work on uncertainty rather than on risk as much as possible, because the numerical probability scale often confounds rather than simplifies what is essential. This point also explains the zigzag structure of Part I of this book. The first big chunk upcoming, Chapters 2 and 3, concerns decision under risk, and the second big chunk, Chapter 4, concerns decision under uncertainty. We nevertheless started in Chapter 1 with decision under uncertainty and not with decision under risk. I did so to encourage 44

2.1 Risk ⊂ uncertainty

45

the readers, from the beginning, to think of decision under risk as being embedded in decision under uncertainty, and never to have thought of decision under risk in isolation. We begin with an introductory example that can be skipped by theoretically oriented (“c”) readers. Example 2.1.1 [Vendor with probabilities given]. Assume Example 1.1.1. Assume that, in addition, there is extensive statistical data available about the weather. One out of four times there will be no rain, one out of four times it will be all rain, and half the times there is some rain. This data is objective and agreed upon by everyone. Writing P for objective probability, we have P(s1) ¼ P(s3) ¼ ¼ and P(s2) ¼ ½. Prospect x (taking ice cream) generates (¼: 400, ½: 100, ¼: 400), meaning that it yields 400 with probability ¼, 100 with probability ½, and 400 with probability ¼. Similarly, y generates (¼:400, ½: 100, ¼: 400), 0 generates (1:0) (also denoted as just 0), and x þ y generates (½:200, ½:0). Note that y generates the same probability distribution over outcomes as x does, assigning the same probability to each outcome. In decision under risk we assume that the preference value of a prospect (i.e., its indifference class) is determined entirely by the probability distribution it generates over the outcomes. Then x and y have the same preference value and are equivalent. For determining preference it then suffices to know the probability distribution generated over the outcomes by a prospect. Hence we often give only that probability distribution. Then x and y are both described by (¼: 400, ½: 100, ¼: 400) and they are equated. The assumption x  y can be violated if utility (defined later) is state dependent. Then the goodness or badness of gaining or losing 400 depends on the weather conditions. We will assume that there is no such dependence. □ For decision under risk, we assume that an objective probability P is given on S, assigning to each event E its probability P(E). Then, with pj denoting P(Ej), each prospect (E1:x1,. . .,En:xn) generates a probability distribution (p1:x1,. . .,pn:xn) over the outcomes, assigning probability pj to each outcome xj. Probability distributions over outcomes taking only finitely many values are called probability-contingent prospects. We now define decision under risk. Assumption 2.1.2 [Decision under risk]. Structural Assumption 1.2.1 (decision under uncertainty) holds. In addition, an objective probability measure P is given on the state space S, assigning to each event E its probability P(E). Different event-contingent prospects that generate the same probability-contingent prospect are preferentially equivalent. □ Because of Assumption 2.1.2, it suffices to describe only the generated probabilitycontingent prospect for determining the preference value of a prospect, without specifying the underlying event-contingent prospect. This will be the approach of the next section. Example 2.1.3 [Decision under risk violated]. Assume a small urn with 20 balls numbered 1 to 20, and a large urn with 200 balls numbered 1 to 200, where from each

46

Chapter 2: EU for risk

a random ball will be drawn and its number inspected. Many subjects prefer gambling on numbers 1–10 from the large urn – for, let us say, $100 – to gambling on number 1 from the small urn (“I have more chances”), even though both event-contingent prospects generate the same probability-contingent prospect (1/20:100, 19/20:0) (Kirkpatrick & Epstein 1992). This implies an empirical violation of Assumption 2.1.2. □ Exercise 2.1.1.a Reconsider Exercise 1.6.7, with Paul and everyone agreeing that the PðEj Þ’s ¼ 1/6 are objective probabilities. Does Assumption 2.1.2 hold for Paul? □ The most commonly used measure on the real axis assigns to each interval [a,b] its length b  a, and to disjoint unions of intervals the sum of their separate lengths. It is called the Lebesgue measure, often denoted l, and we will use it sometimes. On the unit interval [0,1) it is a probability measure. Exercise 2.1.2.a Assume weak ordering, and Assumption 2.1.2. (a) Assume that S ¼ [0,1) and that the probability measure P on S is the Lebesgue measure. Let x be the event-contingent prospect ([0,⅓):3, [⅓,⅔):8, [⅔,1):2). What probability-contingent prospect is generated by x? Let y be the event-contingent prospect ([0,⅓):3, [⅓,⅔):4, [⅔,1):5). What probability-contingent prospect is generated by y? Assume that y ≽ x. Let f be the event-contingent prospect ([0,⅓):8, [⅓,⅔):2, [⅔,1):3). Let g be the event-contingent prospect ([0,⅓):5, [⅓,⅔):4, [⅔,1):3). What is the preference between f and g? (b) Consider the probability-contingent prospect (¾:8,¼:2). Define two different event-contingent prospects that generate it. □ Example 2.1.4 [Regret (Bardsley et al. 2010 Box 3.5)]. Violations of Assumption 2.1.2 may arise because of regret, where you may care about correlations between outcomes (Bell 1982). Assume that S ¼ [0,1) is endowed with the Lebesgue measure as objective probability P, reflecting a random choice of a number from [0,1). Consider a preference ð½0; 1/6 Þ : 10; ½1/6 ; 2/6 Þ : 20; ½2/6 ; 3/6 Þ : 30; ½3/6 ; 4/6 Þ : 40; ½4/6 ; 5/6 Þ : 50; ½5/6 ; 1Þ : 60Þ ≺ ð½0; 1/6 Þ : 60; ½1/6 ; 2/6 Þ : 10; ½2/6 ; 3/6 Þ : 20; ½3/6 ; 4/6 Þ : 30; ½4/6 ; 5/6 Þ : 40; ½5/6 ; 1Þ : 50Þ: Fishburn (1988 p. 273) and Loomes & Sugden (1982 Table 6) argued that this preference can be rational because the regret for an upper choice under the event of s<1/6 will be very big, whereas there is only low regret for the lower prospect under all other events. However, the two event-contingent prospects generate the same probability-contingent prospect, assigning probability 1/6 to each outcome. Hence they must be equivalent under the assumption of decision under risk, and the preference violates decision under risk. As the aforementioned references show, regret leads to violations of transitivity. We will throughout assume Assumption 2.1.2 of decision under risk, and will not consider regret or violations of transitivity. □

2.2 Risk: Basic concepts

2.2

47

Decision under risk: basic concepts

In decision under risk, as in decision under uncertainty, the outcome set is Ñ, with real numbers designating money. As defined in the preceding section, probabilitycontingent prospects, or, briefly, prospects, are probability distributions over Ñ that take only finitely many values.1 The generic notation for a prospect is (p1:x1, . . ., pn: xn), yielding outcome xj with probability pj for each j. Here n is a natural number that can be different for different prospects. Again, we often drop brackets, commas, and colons, writing p1x1. . .pnxn if no confusion with multiplication is likely to arise; another notation is p1x1,p2x2,p3x3. If there are only two outcomes and probabilities, then we often suppress the second probability and write x1p1 x2 instead of (p1:x1, p2: x2), and apb instead of (p:a, 1p:b). For a given prospect, the probabilities pj are the outcome probabilities. It is implicitly understood that outcome probabilities are nonnegative and sum to 1. A prospect x, being a probability distribution over the reals, assigns to each interval in the reals the probability that the outcome of the prospect will fall into that interval. The preference relation over prospects is again denoted by ≽. The figure illustrates the prospect ð3/6 : 20; 1/6 : 14; 2/6 : 12Þ. The circle at the beginning is a chance node. Branches lead to the outcomes. The corresponding probabilities have been indicated. 3

1

2

6

6

6

20 14 12

The next figure illustrates the riskless prospect (1:10). In general, a riskless, or degenerate, prospect yields one fixed outcome with probability 1. It is identified with the outcome. For instance, the outcome 10 is identified with the depicted riskless prospect. 1

10

The following assumption is commonly made for decision under risk, and we make it too. It requires that the underlying state space that generates the randomness is sufficiently rich to generate all probabilities. In particular, S is infinite. Assumption 2.2.1 [Richness for decision under risk]. Every probability distribution over the outcomes that takes only finitely many values is available in the preference domain. □

1

Formally: There is a finite set of outcomes that has probability 1.

48

Chapter 2: EU for risk

Although probabilities are unknown in most practical decision situations, cases of known probabilities are still important. In such cases decision theory can be applied especially fruitfully, in particular if several persons are involved so that agreement and clear communication about probabilities are desirable. Decisions under risk can also serve as a useful benchmark for general decisions under uncertainty, for example when defining attitudes towards ambiguity, as will be explained later.

2.3

Decision under risk as a special case of decision under uncertainty; continued

This is one of several sections in this book that can be skipped without loss of continuity. Such possibilities can be inferred from Appendix K. Hence, in similar cases in future sections this point will not be mentioned again. It is useful to keep in mind that decision under risk is a special case of decision under uncertainty with the underlying state space suppressed for the sake of convenience. The state space describes the physical process that generates the randomness underlying the probabilities. The following example shows that for decision under risk without a state space specified, a state space (we will construct S ¼ [0,1)) can always be defined to serve as underlying state space after all, with respect to which Structural Assumption 1.2.1 (decision under uncertainty) is satisfied. Example 2.3.1. We assume decision under uncertainty with the unit interval [0,1) as (infinite) state space S, as in Example 2.1.4. In what follows, it will be notationally convenient to have S and other events left-closed and right-open (so that disjoint unions generate same kinds of events). We imagine that an arbitrary number will be drawn from S, and the true state s is the number that will be drawn. Event ½3/6 ; 4/6 Þ, for instance, refers to the event that the number drawn will weakly exceed 3/6 but will be strictly below 4/6 . An example of an event-contingent prospect is ð½0; 3/6 Þ : 20; ½3/6 ; 4/6 Þ : 14; ½4/6 ; 1Þ : 12Þ, yielding outcome 20 if the number s drawn is less than 3/6 , yielding outcome 14 if 3/6  s < 4/6 , and yielding outcome 12 if 4/6  s <1. We will henceforth use event-contingent prospects – random variables – defined on intervals, such as x ¼ ([0,q1):x1, [q1,q2):x2, . . . ,[qn1,1):xn), yielding xj if qj1  s < qj, for 0 ¼ q0 < q1 <    < qn ¼ 1. We use the Lebesgue measure l defined in }2.1 as probability measure on S ¼ [0,1). Writing pj ¼ qj  qj1, the prospect x yields outcome x1 with probability p1,. . ., and outcome xn with probability pn. It constitutes a probability distribution p1x1. . .pnxn over the reals, with the pj’s the outcome probabilities. In other words, it generates a probability-contingent prospect. The event-contingent prospect ð½0; 3/6 Þ : 20; ½3/6 ; 4/6 Þ : 14; ½4/6 ; 1Þ : 12Þ, for example, generates the depicted probability-contingent prospect ð3/6 : 20; 1/6 : 14; 2/6 : 12Þ. The event-contingent prospects ð½0; 1/3 Þ : 30; ½1/3 ; 2/3 Þ : 20; ½2/3 ; 1Þ : 10Þ and ð½0; 1/3 Þ : 20; ½1/3 ; 2/3 Þ : 10; ½2/3 ; 1Þ : 30Þ both yield the outcomes 10, 20, or 30, each with probability 1/3 , so

2.3 Risk ⊂ uncertainty continued

49

that both generate the probability-contingent prospect ð1/3 : 10; 1/3 : 20; 1/3 : 30Þ. We assume Assumption 2.1.2. Assumptions 1.2.1 and 2.2.1 are also satisfied. □ The following example illustrates a case where both subjective and objective probabilities are given, and where they must agree. For simplicity, we assume expected value in the example. That the result holds in almost complete generality, also if expected value is not assumed, is shown after. Example 2.3.2 [Objective and subjective probabilities agree under expected value maximization]. Assume Example 2.1.1. Further assume that decision under risk holds, and that the conditions of de Finetti’s Theorem 1.6.1 also hold. Then preferences maximize expected value with respect to a subjective probability measure Q that in principle might be different from P. We show Q ¼ P:

ð2:3:1Þ

Explanation. The prospects (s1:0, s2:100, s3:0) and (s1:100, s2:0, s3:100) generate the same objective probability distribution over outcomes, namely 100½0. Hence, they are equivalent by the assumption of decision under risk. This implies that their subjective expected values, Q(s2)100 and (Q(s1) þ Q(s3))100, must be the same. Hence, Q(s2) ¼ (Q(s1) þ Q(s3)) ¼ ½. The prospects (s1:100, s2:0, s3:0) and (s1:0, s2:0, s3:100) also generate the same objective probability distribution over outcomes, namely (¼:100, ¾:0), and are therefore also equivalent by the assumption of decision under risk. Then Q(s1)100 ¼ Q(s3)100, and Q(s1) ¼ Q(s3) follows. Because these subjective probabilities sum to ½, each must be ¼. We conclude that Eq. (2.3.1) holds. Hence, preferences maximize objective expected value in this example. □ The following exercise shows that, under mild assumptions, subjective probabilities must agree with objective probabilities, also if expected value maximization does not hold. This result will hold for all models considered later in this book. The result implies that the (subjective) probability models developed by de Finetti (1931a), Savage (1954), and others do not deviate from objective probability models, but generalize them by also incorporating situations in which no objective probabilities are given. There have been some misunderstandings about this issue, primarily in the psychological literature. It was sometimes believed, erroneously, that the (“subjective”) expected utility models of Savage (1954) and others were developed to allow subjective probabilities to deviate from objective probabilities (Edwards 1954 pp. 396–397, corrected in Edwards 1962 p. 115; Lopes 1987 p. 258). In modern papers, the term subjective probability is still sometimes used in this unfortunate sense. This misunderstanding may have contributed to the unfortunate separation between risk and uncertainty in some parts of the literature. Mosteller & Nogee (1951 p. 398 and footnote 16) presented a clear and correct discussion. Exercise 2.3.1b [Subjective probabilities must agree with objective probabilities under stochastic dominance]. Assume that S ¼ [0,1) with P the Lebesgue measure j (P([a,b)) ¼ b  a), and assume weak ordering. Define Ej ¼ ½j1 6 ; 6Þ. Consider event-contingent prospects of the form E1x1. . .E6x6, assigning xj to all states s in

50

Chapter 2: EU for risk

j the interval ½j1 6 ; 6Þ. Assume (deviating from Exercise 1.6.7) that decision under risk holds in two ways at the same time for the preference relation ≽: it holds with respect to the objective probability measure P, but also with respect to a, possibly different, subjective probability measure Q. Note that this concerns one and the same preference relation, for which both of these assumptions hold at the same time. Regarding P, the value of a prospect depends only on the probability distribution generated over outcomes through P. For example, the event-contingent prospect 100E10, yielding 100 if the true state is below 1/6 and yielding zero otherwise, is equivalent to the event-contingent prospect 100E60, yielding 100 if the true state is at least 5/6 and yielding 0 otherwise. We further assume a strict stochastic dominance condition with respect to Q: 100A0  100B0 whenever Q(A) > Q(B). This condition holds, for instance, under EV maximization. Write qj ¼ Q(Ej). Show that q1 ¼    ¼ q6 ¼ 1/6 .

Comment.! For any event E with objective probability PðEÞ ¼ 1/6 we can construct a 6-fold partition of S with E one of the elements and all elements of the partition having P-value 1/6 . It follows in the same way as in Exercise 2.3.1 that these events have Q-value 1/6 , too. We can in the same way show that all events with objective probability 1/m must also have subjective probability 1/m. By taking unions we see that all events with objective rational probabilities j/m have the same subjective probabilities j/m. It then readily follows (bounding between rational-probability sub- and supersets) that irrational (in the mathematical sense of being no fraction of integers) objective probabilities must also be identical to subjective probabilities. Hence, for rich event spaces, subjective probabilities must indeed agree with objective probabilities. Further Comment. I briefly discuss situations in which the result of this exercise can be violated. One assumption underlying our analysis is typical of decision theory and is made throughout this book: All prospects are conceivable and the preference relation is complete over all these conceivable prospects. De Finetti, more generally, proved his bookmaking theorem (Theorem 1.6.1) on any linear subspace of prospects. Such cases are important in finance. Financial markets are often incomplete, i.e. it cannot be assumed that all conceivable prospects are available. For example, a prospect 100E0 with E the event that the economy goes down dramatically and every one is losing is not plausible. It may then happen that the result of this exercise does not hold, and risk neutral market (“subjective”) probabilities can deviate from objective statistical probabilities. The deviation may be generated by risk aversion and state dependence of utility (see Example 2.1.1; it will be defined formally later) for instance, with different marginal utilities for different events. This point is central to many works (Kadane & Winkler 1988; Karni 1996; Nau 2003 opening paragraph). The discrepancy between market probabilities and objective probabilities can be used to estimate risk aversion (Bliss & Panigirtzoglou 2004).2 □ 2

Pfanzagl (1968 Theorem 12.6.17) presented further results on the equality of objective and subjective probabilities.

2.4 Decision trees

51

Assignment 2.3.2.b Assume weak ordering, monotonicity, and Assumption 2.1.2 (decision under risk). Assume S ¼ [0,1), with the probability measure P on S the Lebesgue measure. Let x be the event-contingent prospect ([0,⅓):1, [⅓,⅔):2, [⅔,1):3), and let y be the event-contingent prospect ([0,⅓):3, [⅓,⅔):4, [⅔,1):2). Show that y  x. □ 2.4

Choices under risk and decision trees

Fig. 2.4.1a shows two prospects, the upper one yielding $80 with probability 0.5 and $60 with probability 0.5, and the lower one yielding $95 with probability 0.5 and $70 with probability 0.5. The figure displays a choice situation in which you have to choose between the upper and the lower prospect. The square is a decision node; that is, it indicates a situation in which you must choose. Such figures are read from left to right. For instance, Fig. a illustrates a situation where first you arrive at the square node and have to choose if you prefer up or down. If you choose up, you end up in the circle designating the prospect 80½60. If you choose down, you receive 95½70. Fig. b again starts with a decision node where you are to decide which way to go, up or down. Then chance nodes result where chance decides where to go. In Fig. b you have to choose between two prospects each yielding $40 or $20, yielding the higher outcome with probability 0.7 (after up in the decision node) or with probability 0.6 (after down in the decision node). Exercise 2.4.1.a Determine in each of the eight situations depicted in Figure 2.4.1 whether you prefer the upper or the lower prospect. Keep your answers for later use. □ (a)

(b) 0.5

0.5 0.5

0.5

0.7

80 60

0.3 0.6

95 70

(e)

(c)

0.4

0.2

50 × 106 0

20

0.96

60

0.03 0.31

40

1

50 × 106 0

0.69

0 16 0

(h) 0.10

50 × 106

0.89

10 × 106

0.01

0

0.10

0.90 0.11

10 × 106 1

0

4

50

20

10 × 106 0.95

Figure 2.4.1

0.7

0.97

80

(g) 0.04

0.05 1

0.3

40

(f) 0.8

(d)

10 × 106 0.89

50 × 106 0 10 × 106 0

Chapter 2: EU for risk

52

The choice situations in (a), (b), and (c) were trivial. The lower prospect in (a) results from the upper one by replacing the outcome 80 by the outcome 95 and the outcome 60 by the outcome 70. In other words, the lower prospect simply gives better outcomes. It is therefore obvious that the lower prospect is preferable. In (b), the upper prospect results from the lower one by shifting 0.1 probability mass from the bad outcome 20 to the good outcome 40, which again is a definite improvement. In (c), the lower prospect results from the upper one by replacing both outcomes by the worse outcome 50 (or, by shifting all probability mass from the outcomes 80 and 60 to the worse outcome 50) and, hence, it is obvious that the upper prospect is preferable. If you are indifferent, in Fig. 2.4.1e, between the upper and the lower prospect, then your certainty equivalent for the upper prospect is $10  106. It implies that you would equally well like to receive $10  106 for sure as $50  106 with probability 0.8 (and nothing with probability 0.2). Exercise 2.4.2.a Substitute your own certainty equivalent for both prospects depicted in Figure 2.4.2.

(a)

(b) 0.95

0.97

96 ∼

0.05

0

1

4

...

∼

0.03

1

...

0

Figure 2.4.2

□ Further discussion of the choices is postponed until }2.8, although it can be understood at this stage so that interested readers can read }2.8 immediately. Finally, we define risk aversion and risk seeking. Although these concepts will be analyzed only in later chapters, they are defined here to stress their theory-free status. They concern phenomena regarding preferences without any theory assumed. Unfortunately, risk aversion is often confused with concave utility (concepts defined later) in the economic literature. As always in this book, preference without qualification refers to weak preference and not to strict preference. Definition 2.4.1. • Risk aversion holds if every prospect is less preferred than its expected value. • Risk seeking holds if every prospect is preferred to its expected value. • Risk neutrality holds if every prospect is indifferent to its expected value. □

2.5 Utility measurement

53

Risk aversion implies, for example, 50 ≽ 100½0, and risk seeking implies 50 ≼ 100½0. Risk neutrality means that both risk aversion and risk seeking hold, and it is just another name for expected value maximization. Then 50  100½0. It was used in Chapter 1. 2.5

Expected utility and utility measurement

De Finetti’s Theorem 1.6.1 gave a behavioral foundation for expected value maximization for decision under uncertainty. The result can be applied to decision under risk as well. Exercise 2.3.1 and its comment showed that objective and subjective probabilities usually agree. Hence, Theorem 1.6.1, when stated for decision under risk, can serve as a behavioral foundation for expected value maximization. For expected value maximization under risk there are no free subjective parameters in the model. Preferences are uniquely determined and all expected value maximizers behave the same way. With probabilities and outcomes given, we can immediately calculate expected value. The expected value criterion is thus directly observable by itself and does not need a behavioral foundation. It is its own, trivial, behavioral foundation so to speak. In terms of expected utility for uncertainty defined later, expected value for decision under risk concerns the special case where both probabilities and utilities are known, so that nothing about decision attitude remains to be discovered. Table 1.5.4 of finance with bankruptcy at 70K illustrated a case where outcomes were so large that violations of expected value were plausible. The rest of this chapter examines this case for decision under risk. Henceforth, probabilities are known but now utilities will be unknown. The following illustration of a violation of expected value is not very realistic and has a didactical confound (the complexities of infinity). To my surprise, experience has shown that it still works well didactically because students like it. The example is historically important because it led Bernoulli (1738) to develop expected utility. We will return to the example with Bernoulli’s explanation later. Example 2.5.1 [St. Petersburg paradox]. Consider the following game. A fair coin will be flipped until the first heads shows up. If heads shows up at the kth flip, then you receive $2k. Thus, immediate heads gives only $2, and after each tails the amount doubles. After 19 tails you are sure to be a millionaire. Think for yourself how much it would be worth to you to play this game. The expected value of the game is 1/2

 2 þ 1/4  4 þ 1/8  8 þ 1/16  16 þ    ¼ 1 þ 1 þ 1 þ 1 þ    ¼ 1:

Thus if you maximize expected value, then this game is worth more to you than any amount of money. In reality, people pay considerably less to participate in the game, something like $5 (as Bernoulli wrote based on casual observations), showing that expected value does not hold empirically when large amounts are involved. □

Chapter 2: EU for risk

54

In Chapter 1 we first introduced a number of plausible preference conditions. We then came to the, possibly surprising, conclusion that a specific preference model was implied, namely EV. }1.8 further discussed general features of such behavioral foundations. We could present all behavioral foundations in the same spirit, especially those with a normative status. For brevity’s sake, we will not do so. We follow a different system henceforth, as stated in the Introduction. It is shorter and can be applied both normatively and descriptively, and is as follows. First an empirical measurement of the subjective parameters is described that relates them directly to preferences. Next, consistency of measurement gives a behavioral foundation. We summarize the structural assumption needed for the following analysis. It is implied by Assumptions 1.2.1 (decision under uncertainty), 2.1.2 (decision under risk), and 2.2.1 (richness), as can be seen. Structural Assumption 2.5.2 [Decision under risk and richness]. ≽ is a preference relation over the set of all (probability-contingent) prospects, i.e., over all finite probability distributions over the outcome set Ñ. □ Definition 2.5.3. Under Structural Assumption 2.5.2, expected utility (EU) holds if there exists a strictly increasing function U, the utility (function), from the outcome set to Ñ, such that the evaluation depicted in Figure 2.5.1 represents preferences. □ p1

x1 .

p1U(x1) + …+ pnU(xn)

. . pn

the expected utility (EU ) of the prospect

xn

Figure 2.5.1

Under EU, the utility function is the subjective parameter characterizing the decision maker. The following exercise shows how to derive decisions from the EU model. Its elaboration is immediately given in the main text. Exercise 2.5.1.a Assume EU with U(0) ¼ 0, U(100) ¼ 1, U(81) ¼ 0.9. Determine the preference between 1000.90 and 1000.181. □ In Exercise 2.5.1 EU(1000.90) ¼ 0.90  U(100) þ 0.10  U(0) ¼ 0.90; EU(1000.181) ¼ 0.10  U(100) þ 0.90  U(81) ¼ 0.10 þ 0.81 ¼ 0.91. Hence 1000.181 has the higher EU and is preferred. pffiffiffi Exercise 2.5.2.a Assume EU with UðaÞ ¼ a. Determine the preference between 490.916 and 810.716. □

2.5 Utility measurement

55

Example 2.5.4 [St. Petersburg paradox explained using EU]. Consider Example 2.5.1. Bernoulli argued that expected utility with a logarithmic utility function is plausible. Then the expected utility of the game is X1 X1 j j 2 lnð2 Þ ¼ j 2j lnð2Þ ¼ 2lnð2Þ ¼ lnð4Þ:3 j¼1 j¼1 Hence the certainty equivalent is $4, fitting Bernoulli’s empirical claims well. □ As in }1.3, we can use the elicitation method, and derive properties of utility from observed choices. Part (a) of the following exercise, and what follows, illustrate this point. The exercise illustrates the applicability of expected utility, with some choices used to assess the model and this assessment subsequently used to predict new choices. Its elaboration is given immediately following the exercise. Exercise 2.5.3.a Assume that you observe some preferences of an agent. You know that the agent satisfies EU, and can set U(0) ¼ 0 and U(100) ¼ 1. (a) Assume that 50  1000.580 for the agent. What is U(50)? (b) Under the indifference in part (a), you can predict the agent’s preference between (0.40:100, 0.20:50, 0.40:0) and (0.33:100, 0.33:50, 0.34:0). What is it? Elaboration. (a) Because of the equivalence, U(50) ¼ EU(1000.580) ¼ 0.58  1 þ 0.42  0 ¼ 0.58. (b) EU(0.40:100, 0.20:50, 0.40:0) ¼ 0.40  1 þ 0.20  0.58 þ 0.40  0 ¼ 0.40 þ 0.116 þ 0 ¼ 0.516. EU(0.33:100, 0.33:50, 0.34:0) ¼ 0.33  1 þ 0.33  0.58 þ 0.34  0 ¼ 0.33 þ 0.191 þ 0 ¼ 0.521. The latter prospect has the higher EU and is preferred. □ Exercise 2.5.4.a Assume EU with U(0) ¼ 0 and U(100) ¼ 1. (a) Assume that 60  1000.700 for the agent. What is U(60)? (b) Under the indifference in part (a), what is the preference between 600.700 and 1000.490? □ The best way to learn about the general way to measure utility is to invent it by yourself. You can do so in the following exercise. Its elaboration is given next in the main text. Exercise 2.5.5.a Assume EU with U(0) ¼ 0 and U(100) ¼ 1. What kind of preferences should you observe to find out what U(30) and U(70) are? □

3

P1 j P1 j P1 P j ðj1Þ j 1 P1 Second}=} ¼ 1 þ 1=2 1 j¼1 j2 : P1 : j j¼1 j2 ¼ j¼1 2 þ j¼1 ðj  1Þ2 ¼ 1 þ =2 j¼2 ðj  1Þ2 Then j2 ¼ 2. j¼1

Chapter 2: EU for risk

56

1 U 0.8 0.40

0.80

100

30 ∼

100

70 ∼ 0.60

0

0.4 0.20

0

0

0

30

70

100 $

Figure 2.5.2 Two indifferences and the resulting U curve.

Measuring utility. To analyze risky decisions, we want to measure utility U. Figure 2.5.2 shows an example. Assume EU with the normalization U(0) ¼ 0 and U(100) ¼ 1. The first indifference implies Uð30Þ ¼ 0:40  Uð100Þ þ 0:60  Uð0Þ ¼ 0:40: The second indifference implies Uð70Þ ¼ 0:80  Uð100Þ þ 0:20  Uð0Þ ¼ 0:80: The graph in Figure 2.5.2 then results from linear interpolation. In general, assume two fixed outcomes M > m, and assume that we have normalized U(m) ¼ 0 and U(M) ¼ 1. (In Figure 2.5.2, m ¼ 0 and M ¼ 100.) We will see later (Exercise 2.6.4) that we may always assume such normalized utility values. For each outcome a such that M  a  m, we can elicit the standard-gamble (SG) probability with respect to m and M, being the probability p such that the equivalence in Figure 2.5.3 holds. Applying EU gives UðaÞ ¼ pUðMÞ þ ð1  pÞUðmÞ ¼ p ½the SG Eq:

ð2:5:1Þ

In other words, U(a) is simply the SG probability p. Exercises 2.5.3a and 2.5.4a illustrated this procedure.

p

M

α ∼ 1−p

m

Figure 2.5.3 The SG probability p of a.

2.6 Behavioral foundation of EU

57

The SG method directly relates utility to decisions, in a very simple manner. The method shows how to make exchanges between outcomes and probability. This way outcomes and probabilities become commensurable. Through a SG consistency condition defined later, this commensurability will capture the essence of EU. Surveys of utility measurement under EU are in Farquhar (1984) and Seidl & Traub (1999). Assignment 2.5.6.c This assignment does not concern binary choice, as does most of this book, but it concerns choosing from a continuum of prospects. ½

½

1.5β

−β

(a) Assume that you can invest an amount b at your discretion in a stock that with ½ probability yields a return of 2.5b and, thus, a profit of 1.5b. There is a probability ½ of ruin, in which case you lose your investment b. You maximize EU, with utility function U(a) ¼ 1  ea. How much do you invest? What are the resulting EU and CE? (b) The same question, but now U(a) ¼ 1  eya for a general y > 0. (Hint: substitute ya for a in part (a)). □ 2.6

Consistency of measurement and a behavioral foundation of expected utility

We assume in this chapter, as we do throughout this book, that the outcome set is Ñ. It can be seen, though, that the behavioral foundation of expected utility presented in Theorem 2.6.3, and the analysis of this whole chapter, hold for general outcome sets, and that we never need the outcome set to be Ñ. The only change required for general outcomes is that inequalities between outcomes such as M > m be replaced by preferences between outcomes such as M  m. A necessary condition for EU to hold, and for the measurement of utility to work well, is that a probability p to solve the indifference in Figure 2.5.3 exists for all M > a > m. This requirement is called standard-gamble (SG) solvability. It is the only continuity-like condition needed in the analysis to follow. It entails that the probability scale is at least as refined as the outcome scale. Solvability conditions are less restrictive and better capture the empirical meaning of a model than continuity (which essentially needs connectedness of the reals in preference foundations; Luce et al. 1990 p. 49 l. 10; Wakker 1988). Another immediate implication of EU, based on utility being strictly increasing, is as follows.

58

Chapter 2: EU for risk

Standard-gamble (SG) dominance. For all outcomes M > m and probabilities p > q: p

M

q

M

· 1−p

m

1−q

m

Moving positive probability mass from an outcome to a strictly better outcome entails a strict improvement. The condition is a weak version of stochastic dominance (defined later). Exercise 2.6.1.a Assume EU. Verify, for general U(M) > U(m), that p ¼ (U(a) U(m))/(U(M)U(m)) in Figure 2.5.3. □ Exercise 2.6.2.b Show that SG dominance holds under EU. In other words, assume a preference relation over prospects that can be represented by EU. Show that the preference relation satisfies SG dominance. □ Exercise 2.6.3.b Assume weak ordering, SG dominance, and SG solvability. Show, for M > m, that for the SG probability p in Figure 2.5.3: (a) (b) (c) (d)

p is uniquely determined. If a  m, then p ¼ 0. If a  M, then p ¼ 1. If m < a < M, then 0 < p < 1. □

The following exercise deals with an important property of the utility function in expected utility. If you have no clue how to resolve the exercise, then a comment similar to the one preceding Exercise 1.3.4 applies. You probably have not yet fully internalized the conceptual meaning of decision models. A rethinking of Definition 2.5.3 and Exercise 2.5.3, and a rereading of some of the preceding text, may then be useful. Alchian (1953) gives a didactic and detailed account of the empirical meaning of utility and of the following exercise. Exercise 2.6.4!a [Free level and unit of U]. Show that, if EU holds with a utility function U – i.e., the expectations of U as in Figure 2.5.1 represent preferences – then so does EU with utility function U* ¼ t þ sU, where t 2 Ñ and s > 0. □ Exercise 2.6.4 shows that we can normalize the utility scale as we like, and choose where U will be 0 and 1. Exercise 2.6.5.c Assume EU. Show that U is continuous if and only if there exists a certainty equivalent for each prospect. □ The implications of EU just described are relatively uncontroversial. We now turn to the most interesting, and most controversial, implication – the well-known independence condition. We consider here a version called standard gamble consistency,

2.6 Behavioral foundation of EU

59

which states the condition in terms of consistency of utility measurement. Most normative discussions of EU concern the independence condition. Empirical tests of EU and its alternatives usually examine how independence is violated, and most nonexpected utility models weaken independence in specific ways. Such alternative models will be presented later. In a perfect world where EU would hold perfectly well, the utility measurement through the SG probability in Figure 2.5.3 would provide exact measurements of U that would give perfect predictions. Unfortunately, there usually are measurement errors and biases, and models do not hold perfectly well. McCord & de Neufville (1986) criticized the SG method for being particularly prone to empirical biases. The main reason is that this method involves a riskless outcome a, and such outcomes are often evaluated in deviating manners, generating violations of EU. It may, therefore, be safer to avoid their use. McCord & de Neufville, and several other authors (Carthy et al. 1999 }2; Davidson, Suppes, & Siegel 1957; Officer & Halter 1968 p. 259), recommended embedding SG indifferences as in Figure 2.5.3 into more complex risky choice prospects. To prepare for this embedding, we discuss the (probabilistic) mixing of prospects such as xly, clarified in the next example. Example 2.6.1 [Probabilistic mixing of prospects]. Consider the left-hand two-stage prospect in Figure 2.6.1. In the first chance node there is a ½ probability of going up and a ½ probability of going down. If up, the prospect 2001/30 results. These ⅓ and ⅔ probabilities are conditional probabilities, being conditional on the event of going up. The other probabilities are similar. The probability of receiving $200 is, therefore, 1/2  1/3 ¼ 1/6 , the probability of $100 is also 1/6 , and the probability of 0 is 1/2  2/3 þ 1/2  2/3 ¼ 2/3 . That is, the probability distribution is the same as in the right-hand prospect. ⅓ ½

200 6

⅔ ⅓ ½ ⅔

200

0 6

100

⅔

100

0

0

Figure 2.6.1

□ Definition 2.6.2. For general prospects x, y, and 0  l  1, the mixture depicted in Fig. 2.6.2a, also denoted xly, is the prospect that assigns, to each outcome a, l times the probability of a under x plus 1l times the probability of a under y. It is called a (probabilistic) mixture of x and y. □

60

Chapter 2: EU for risk (a) λ

x y

1−λ (b)

⅓

200 0

⅔ (c)

⅓

100 0

⅔

Figure 2.6.2

In Figure 2.6.1, x was the prospect in Fig. 2.6.2b, y was the one in Fig. 2.6.2c, and l was ½. The mixture xly can be thought of as resulting from a two-stage procedure, where in the first stage prospect x results with probability l and prospect y with probability 1l, and in the second stage the prospect resulting from the first stage (x or y, as the case may be) is played out.4 Figure 2.6.3 shows another example. ¼

8

2

2

x= ¼

5

y=

5 ;

;

(λ = 4 5 )

2

0

2

¼ 4

5

8

2

5

8

2

5

5 ¼

The mixture x4/5y can be depicted as

0 and is equal to

2

5

10 5

5

2 0

2 2

Figure 2.6.3 4

This footnote is meant to be read only by specialists in decision theory. It concerns the reduction of compound lotteries (¼ prospects) assumption. If mixtures of prospects are interpreted as physically resulting from a twostage procedure as just described, with physical time proceeding, then the reduction of compound prospects assumption is used, which entails that probabilities of subsequent chance nodes are multiplied. In our analysis, the only objects considered are one-stage prospects. Two-stage illustrations are only a didactic device for depicting one-stage prospects.

2.6 Behavioral foundation of EU

61

I next discuss linearity of EU in probabilities. Because this property is important for understanding EU, I shall explain it in some detail. We first consider this property for general integrals. If with probability ½ you receive a prospect with expected return $100, and with probability ½ you receive a prospect with expected return $200, then your expected return is ½  100 þ ½  200 ¼ 150. If you receive a medical treatment where: • with probability ¼ people live for another 40 years on average, • with probability ¾ people live for another 4 years on average, then on average you live for another ¼  40 þ ¾  4 ¼ 10 þ 3 ¼ 13 years. In general, if prospect x has an expectation (of money, life duration, or whatever) EV(x) and prospect y has an expectation EV(y), then receiving x with probability l and y with probability 1l has expectation lEV(x) þ (1l)EV(y). This means that expectation is linear with respect to probabilistic mixing. The result holds in particular for EU, that is, not only if the expectation concerns money or life years, but also if the expectation concerns utility. If x has expected utility EU(x), and y has expected utility EU(y), then receiving x with probability l and y with probability 1l has expected utility lEU(x) þ (1l)EU(y). Exercise 2.6.6.a Show that EU is linear in probability; i.e., EU(xly) ¼ lEU(x) þ (1l)EU(y). □ We now return to the idea of McCord & de Neufville (1986). They proposed to consider equivalences such as in Figure 2.6.4, rather than the SG equivalences of Figure 2.5.3. They called the resulting measurement method the lottery-equivalent method. Basically, the same indifference is elicited as in Figure 2.5.3, but now a and the equivalent SG-prospect have been mixed with another “common” prospect C. This way they avoid the use of riskless prospects. p λ

α ∼

1–λ

C

M

λ 1–p 1–λ

m

C

Figure 2.6.4 The lottery-equivalent method of McCord & de Neufville (1986) (l > 0).

Because of the linearity in probability of EU (Exercise 2.6.6), the EU of the left prospect in the figure is lU(a) þ (1l)EU(C), and of the right prospect it is l(pU(M) þ (1p)U(m)) þ (1l)EU(C). The equality of these expected utilities implies the same equality U(a) ¼ pU(M) þ (1p)U(m) ¼ p as resulting from the SG equivalence of the preceding section (the SG Eq. (2.5.1)).5 5

Cancel the common term (1l)EU(C), divide by the positive l, and set again U(m) ¼ 0 and U(M) ¼ 1.

62

Chapter 2: EU for risk

In a perfect world where EU would hold perfectly well, there would be no inconsistencies in observations. Then it would not matter if we use Figure 2.5.3 or Figure 2.6.4 to measure p, and p should be the same in both figures. This identity requirement is called standard-gamble (SG) consistency. Then the indifference in Figure 2.5.3 should always imply the indifference in Figure 2.6.4. This implication is stated formally in Figure 2.6.5. p SG consistence holds if p

λ

λ

α

M

α ∼

1–p

∼

implies 1–p

M

m 1–λ

C

1–λ

m

C

for all outcomes α, M, m, all probabilities p and λ, and all prospects C.

Figure 2.6.5

The condition has normative appeal. If the left indifference in Figure 2.6.5 – i.e., the indifference in Figure 2.5.3 – holds, then in the right indifference (Figure 2.6.4) you receive the same C for both prospects if going down (1l), and equivalent things if going up (l), so that you are equally well off in either case. Hence, the two compound prospects seem to be equally attractive to you. The following theorem shows that SG consistency, a weak version of the well-known independence condition (see next section) together with some natural conditions, is not only necessary, but also sufficient, for EU. Von Neumann & Morgenstern (1947) presented a first version of this theorem, but their result was incomplete. Several authors have subsequently contributed to variations of the result (Fishburn 1970 p. 103). The following theorem is yet another variation. Theorem 2.6.3 [EU for risk]. Under Structural Assumption 2.5.2 (decision under risk and richness), the following two statements are equivalent. (i) Expected utility holds. (ii) ≽ satisfies: • weak ordering; • SG solvability; • SG dominance; • SG consistency. □ We saw, in Exercise 2.6.4, that the utility U can be replaced by any U* ¼ t þ sU for real t and positive s. Here s is the unit parameter because it determines the unit of measurement and t is the level parameter because it affects absolute levels of utility. It turns out that no other utility functions are possible than the ones just described.

2.7 Independence

63

In general, a function in a particular model is unique up to unit and level if it can be replaced by another function if and only if the other function differs by unit and level. We also call such functions interval scales. The term cardinal has also been used to refer to this mathematical property, but this term has many interpretational connotations. We therefore will not use it. Observation 2.6.30 [Uniqueness result for Theorem 2.6.3]. In EU, utility is unique up to level and unit. □ Comment 2.6.4 [Organization of measurements and behavioral foundations; the five-step presentation]. The line of presentation leading to Theorem 2.6.3 is typical of the presentation of decision models throughout this book. It involved the following five steps: (1) The model, EU, was defined in Definition 2.5.3, with utility as subjective parameter; (2) Exercises 2.5.1 and 2.5.2 showed how to derive decisions from the model’s parameters; (3) Exercises 2.3–2.5 and Eq. (2.5.1) presented the elicitation method for deriving the model’s parameters from decisions; (4) }2.6 presented consistency conditions for measurements; (5) the consistency conditions for measurements gave a behavioral foundation in Theorem 2.6.3. □ Comment 2.6.5 [Variance of utility]. It may be surprising that only the probability-weighted average of utility plays a role under EU, and no other aspects of the utility distribution. Receiving a utility of 0.6 for sure is equated with receiving utility 1 with probability 0.6 and utility 0 with probability 0.4. It may seem that the risk and, for instance, the variance of the utility received should also be relevant and that they are inappropriately ignored here (Gandjour 2008 p. 1210; Rode et al. 1999 p. 271). The key to understanding why this is not so, and to understanding the essence of utility, lies in the SG Figure 2.5.3. Because utility is measured in this way, essentially in terms of probabilities, it becomes commensurable with probability and risk. The critical condition of EU, SG consistency, implies that the exchange rate between value and risk as measured in the SG equivalence is universally valid, also within complex decision trees. In this manner, all of probability and risk that is relevant for the preference value of prospects is captured by utility and its average. □ 2.7

Independence and other preference conditions for expected utility

In the literature, many variations have been used of the preference conditions presented in the preceding section. Somewhat stronger (more restrictive) than SG consistency is the substitution principle: λ

λ

x

x ∼ y implies

y

∼ 1−λ

C

· 1−λ

C

Chapter 2: EU for risk

64

It entails that replacing a prospect in a mixture by another equally good prospect does not affect the preference value of the mixture. SG consistency concerned the special case where x was a sure outcome and y a two-outcome prospect. Yet stronger is the independence condition, a condition most often used in the literature for behavioral foundations of EU: λ x

x

λ

y ·

y implies 1−λ

C

1−λ

C

It entails that improving a prospect in a mixture improves the mixture. Substitution follows by first applying independence with x ≽ y and then with x ≼ y. A violation of independence suggests a violation of a conservation-of-preference principle, alluded to before when discussing Dutch books. Although I will not try to formalize this general idea, the following reasoning may be suggestive: If the preference between the mixtures were reversed so that replacing x by y in the mixture would lead to a strictly preferred prospect, then we would have created positive preference value by keeping C fixed and worsening x. It signals that there must be some positive interaction between C and y (or negative between C and x). Given that in no possible physical world C and x, or C and y, can coexist (you never receive both C and x, or both C and y), the interaction must be purely psychological, occurring only in the mind of the decision maker. The decision maker is, magically, creating value out of nothing but own imagination. Independence, and its weakenings presented here, may seem to be completely self-evident at first sight, and I also think they are normatively compelling. However, these conditions have many implications, and many people, upon examining the implications, come to reject independence as a universal normative principle. As the psychologist Duncan Luce once put it (Luce 2000 }2.4.2), “Although this line of rational argument seems fairly compelling in the abstract, it loses its force in some concrete situations.” We will return to criticisms of independence in later sections. Because SG consistency is weaker than independence, the implication of (ii) ) (i) in Theorem 2.6.3 (EU for risk) with SG consistency is a stronger result than it would be if independence were to replace SG consistency. The result with independence follows as an immediate corollary of the result with SG consistency and this does not hold the other way around. In this sense, SG consistency is a more useful preference condition. In many applications, the implication (i) ) (ii) in Theorem 2.6.3 is useful. Strong preference conditions may be interesting in such applications. For example, if we want to falsify EU empirically, then it will be easier to obtain violations of independence than of SG consistency. Because independence is stronger than SG consistency, violations of the former are easier to detect than violations of the latter. For example, if there are no indifferences in our data then SG consistency cannot be falsified in any direct manner, but independence can be. This will happen, for

2.8 Basic choice inconsistencies

65

instance, when we discuss the Allais paradox in }4.12. This paradox directly falsifies independence, showing that EU is violated, but it does not directly falsify SG consistency. Hence, it is useful to know various preference conditions for decision theories, and not only the weakest ones that give the most general behavioral foundations. In the presence of the other conditions for EU, SG dominance implies a very natural condition, usually considered a hallmark of rationality, namely (first-order) stochastic dominance: shifting probability mass from an outcome to a preferred outcome should lead to a preferred prospect. In other words, the more money the better. This preference condition extends SG dominance to general prospects with possibly more than two outcomes, using weak (unqualified) rather than strict preference. As will be explained later, the condition defined here is equivalent to other formulations used in the literature, stated in terms of distribution functions or improvements of outcomes. Exercise 2.7.1.!a Show that EU implies stochastic dominance. □ Exercise 2.7.2.b Use Exercise 2.6.6 to show that EU implies independence. □ The following variation of independence is similar to Savage’s (1954) sure-thing principle, requiring that preference be independent of common outcomes, and defined later for uncertainty (Eq. (4.8.1)). We call it the sure-thing principle for risk, and it is defined in Figure 2.7.1. It can be seen that independence and this condition are, in general, logically independent. In the presence of natural conditions plus a probabilistic continuity condition they become equivalent. Hence, the sure-thing principle for risk could also be used to obtain a preference foundation for EU. We do not elaborate on this point. p β p2 x2

p q2

. .

pm

. xm

. .

qn

p γ p2 x2

β y2

yn

q2

pm

. xm

γ y2

. .

. .

implies

.

p

qn

. yn

Figure 2.7.1 The sure-thing principle for risk.

Exercise 2.7.3.!a Show that EU implies the sure-thing principle for risk. □ 2.8

Basic choice inconsistencies

The following exercise will be discussed immediately in the text that follows. Exercise 2.8.1.a Substitute your own certainty equivalent for both prospects depicted in Figure 2.8.1.

Chapter 2: EU for risk

66 (a)

(b) 0.95

0.31

96 1 ∼

0.05

24

16 1

...

∼ 0.69

...

0

Figure 2.8.1

□ Violations of EU will be discussed in later chapters, and the choices in Figure 2.4.1 will then play a role. This section only discusses some violations of general choice principles, revealed by the choices in Figures 2.4.1, 2.4.2, and 2.8.1. The choice in Fig. 2.4.1d can be related to Figs. 2.4.2b and 2.8.1b, where the two prospects reappear. It is natural that the prospect that you preferred in Fig. 2.4.1d had the highest certainty equivalent. If this is not the case for you, then a violation of stochastic dominance or transitivity can be derived from your choices. Empirically, a majority of subjects exhibit such a violation. That is, they choose the upper prospect in Fig. 2.4.1d but assign a lower CE to it in Fig. 2.4.2b relative to Fig. 2.8.1b. These violations are known as preference reversals (Bardsley et al. 2010 Box 3.4). The certainty equivalents in Figs. 2.4.2a and Fig. 2.8.1a can also be related to each other. The prospect in Fig. 2.8.1a dominates the one in Fig. 2.4.2a because the 0 outcome has been replaced by the better outcome 24. Hence, the prospect in Fig. 2.8.1a should be preferred to the one in Fig. 2.4.2a. It should, therefore, have a higher certainty equivalent. If your certainty equivalent in Fig. 2.8.1a was lower than in Fig. 2.4.2a, then a violation of stochastic dominance or transitivity can be derived from your choices. Empirically, a majority of subjects exhibit such a violation. (You were exposed to about the most tricky stimuli that this field has developed.) This choice anomaly was first discovered by Birnbaum et al. (1992). Their explanation is that zero outcomes generate special biases, with people sometimes overweighting their role but on other occasions, such as here, improperly ignoring them. Empirical findings such as those just discussed are notorious because they reveal many violations of rationality requirements, such as transitivity or stochastic dominance. The prevalence of such anomalies has led some authors to conclude that studying individual preference is a hopeless task because, as they argue, such anomalies clearly show that we are not measuring “true values” in any sense, and we are only measuring random choices. Such views are captured in pessimistic versions of the constructive view of preference, discussed by Slovic (1995). There remain, fortunately, enough regularities in data to make the empirical study of decision theory worthwhile. In addition, for many decisions, individual preferences and utilities are the crucial and decisive factors. We then have to measure them as well as we can, no matter what the difficulties are. Optimistic versions of the constructive view of preference argue that we, decision analysts, aware of the deficiencies of our

2.9 Proof

67

measurement instruments, should work and interact more with clients, “not as archaeologists, carefully uncovering what is there, but as architects, working to build a defensible expression of value” (Gregory, Lichtenstein, & Slovic 1993 p. 179). Using interactive sessions we can get more out of fewer subjects. Whereas it is common jargon in the field to refer to choice anomalies as irrationalities and biases on the part of the decision makers, it may be argued that these anomalies are a problem of our measurement instruments rather than of the decision makers. The purported biases and irrationalities are usually discovered in experimental measurements of preference values. Such measurements, especially if based on hypothetical choice – as done in Figures 2.4.1, 2.4.2, and 2.8.1 – are hard to relate to for subjects. When answering the questions concerning these figures, you may have felt that your answers were at least in part random. Thus our measurement instruments, even if the best conceivable, do not perfectly tap into the value systems of subjects. In measurements of preference and value it is, therefore, desirable to clarify the nature of stimuli to subjects as much as possible, and to use real incentives in descriptive studies whenever possible. For prescriptive purposes, a big value of decision theory lies in uncovering anomalies such as those just described. Such anomalies do exist in individual choices and indeed cannot reflect true values. By revealing such anomalies, decision theory can show people that in some situations they are not following their true value system, and that they can and should improve their decisions in such situations. In other words, in such situations decision theory can bring new insights and improve decisions. It can do this by using simple consistency checks, without any need to know the exact true values of a client. As pointed out by Raiffa (1961), it is not problematic but, rather, essential for prescriptive applications of decision theory that violations of the rational models exist in natural behavior. Appendix 2.9

Proof of Theorem 2.6.3 and Observation 2.6.30

The proof of Theorem 2.6.3 is simple. It makes a crucial achievement of EU transparent, namely that EU provides commensurability between outcomes and probability. SG measurements settle the exchange rate between outcomes and probabilities, and this exchange rate should subsequently apply to all situations. This is the idea underlying the proof. It was explained in the main text that the preference conditions are necessary for EU. We, henceforth, assume the preference conditions and derive EU. Take two outcomes M > m. Define U(m) ¼ 0 and U(M) ¼ 1 for now. For each m < a < M define U(a) as the SG probability of a (Figure 2.5.3), which can be done because of SG solvability. Consider a prospect with all outcomes between m and M. Figure 2.9.1 is explained next. The first equivalence follows from SG consistency, by replacing the riskless prospect xj by its equivalent standard gamble. The second equivalence follows the same way, by repeated application. The equality is by multiplication of probabilities. Every prospect is thus equivalent to the prospect that with a probability equal to the EU of the original prospect yields outcome M, and outcome m otherwise. By SG

Chapter 2: EU for risk

68

p1

x1

p1

. . .

. . . pj

xj ∼

x1

U(xj)

pj

. . .

. . . pn

xn

p1

pn

1−U(xj)

U(x1)

. m . 1−U(x1) . M

U(xj)

∼

pj

pn 1−U(xn)

p1U(x1) + ⋅⋅⋅ + pnU(xn)

M

m . . 1−U(xj) .

m

U(xn)

xn

M

M

= 1−p1U(x1) − ⋅⋅⋅ − pnU(xn)

m

M m

Figure 2.9.1

dominance, the latter prospects, having only m and M as outcomes, are ordered by their EU. By transitivity, so are all prospects with outcomes between m and M. We have proved that EU holds on the domain of those prospects. For the uniqueness result in Observation 2.6.30 applied to the domain of all prospects with outcomes between m and M, we can freely choose unit and level of U by Exercise 2.6.4. This is the only freedom we have. For any alternative U*, we can substitute U*(m) ¼ t and U*(M) ¼ s þ t for any real t and s > 0, after which U* is uniquely determined from SG equivalences as t þ sU for the utility U with U(m) ¼ 0 and U(M) ¼ 1. We can carry out the above reasoning for any m < M, thus obtaining EU on every bounded part of our domain and, finally, obtain EU on the whole domain by routine extensions. For the extensions of the domain (reducing m and enlarging M) it is easiest to rescale all newly defined models such that utility is 0 at the original m we started with and it is 1 at the original M we started with. Then these models all coincide on common domains by the uniqueness result just established, and they are genuine extensions of each other. Observation 2.6.30 thus holds on the domain limited by any m and M and, consequently, on the whole domain. By SG dominance with p ¼ 1 and q ¼ 0, a > b implies a  b. This implies that U increases strictly. The proof is now complete. The proof presented here adapts some appealing substitution-based proofs in the literature (Arrow 1951a pp. 424–425; Luce & Raiffa 1957 pp. 27–28) to our preference conditions. I constructed the proof of the bookmaking theorem in Appendix 1.11 by using the proof of this appendix and then applying a duality between states and outcomes (Abdellaoui & Wakker 2005 end of }1). Figure 1.11.1 is the analog of Figure 2.9.1. In the former figure we replaced every event E by its certainty equivalent P(E) and then used linearity in outcomes. In the latter figure we replaced every outcome by its standard-gamble probability (of getting M versus m) and then used linearity in probability. □

3

Applications of expected utility for risk

Given the many applications of expected utility for decision under risk, we dedicate a separate chapter to this topic. Throughout this chapter we make the following assumption, often without further mention. It implies Structural Assumption 2.5.2 (decision under risk and richness), adding the assumption of EU. Structural Assumption 3.0.1 [Decision under risk and EU]. ≽ is a preference relation over the set of all (probability-contingent) prospects, which is the set of all finite probability distributions over the outcome set Ñ. Expected utility holds with a utility function U that is continuous and strictly increasing. □ The assumption that all finite probability distributions are available in the preference domain entails, in fact, a strong richness restriction, similar to our assumption that all real-valued outcomes are available in the domain. The assumption is, however, commonly made in the literature on decision under risk and it facilitates the analysis, which is why we use it too.

3.1

An application from the health domain: decision tree analysis

My experience with applications of decision theory mostly come from the medical domain. Although discussions of medical examples can at times be depressing, dealing with human suffering, the medical domain is one of the most important fields of application for decision theory. Hence, I present a medical application. We consider a simplified decision analysis for patients with laryngeal cancer in stage T3 (a particular medical state of the cancer with no metastases; McNeil et al. 1981). In this subsection, as an exception, outcomes are nonmonetary. We, accordingly, drop the assumptions that U be continuous or strictly increasing. The choice between surgery and radiotherapy is difficult for stage T3, and there are no clear medical indications favoring one treatment over the other. The advantage of surgery is that there is a smaller chance of recurrence, so that the patient has a better chance of survival. The disadvantage is, however, that the voice is lost and the patient will have to live with artificial speech. The best treatment decision crucially depends on the individual subjective preferences of the patient. For a teacher, the loss of the 69

70

Chapter 3: Applications of EU

voice may be a serious loss and the teacher may, therefore, prefer radiotherapy. A piano player may prefer surgery. At any rate, it is not possible to prescribe a general treatment that would be optimal for all patients. It is essential for the best treatment to consider the subjective value system of the patient. Inputs from the patient should play an explicit role in the decision process. Giving formal status to patient inputs is an important contribution of decision theory to the health domain. Up until the early 1980s, no technique was available in the medical domain to incorporate subjective individual preferences of patients in a systematic way. An often-used criterion was the five-year survival rate, describing how many patients were still alive five years after treatment. A breakthrough was achieved when McNeil et al. (1978, 1981) demonstrated how the techniques of decision theory can be used for medical decisions. It led to a revolution in the medical domain, where quality of life and inputs from the patient became generally recognized as essential components of optimal treatments. 0.6 Cure Radiotherapy

Recurrency, surgery 0.4

Normal voice 0.4

Artificial speech

0.6 0.7 Cure

Artificial speech

Surgery 0.3 Recurrency 0.3 : death

Artificial speech

0.7

Figure 3.1.1 Choice between radiotherapy or surgery for a patient with larynx-cancer (stage T3).

Figure 3.1.1 depicts the risks and the events of the situation considered here. In an EU analysis, we should not only determine the probabilities of risks, but also the utilities of outcomes. In the health domain, the probabilities are usually provided by epidemiologists, based on large studies of the literature and of medical data. The probabilities in Figure 3.1.1 have been obtained in this manner. We will not discuss them further. To determine the utilities of the patient, the standard gamble (SG) method can be used. The best outcome in the analysis, normal voice, is assigned utility 1 and the worst outcome, death, is assigned utility 0. Next the SG probability p of life with artificial speech with respect to normal voice and death is determined (Figure 3.1.2, which is the analog of the SG Figure 2.5.3). Needless to say that such measurements need extensive psychological preparations,

3.2 Risk aversion p Artificial speech

71

Normal voice

∼ 1−p

Figure 3.1.2 The SG question: For which p is the gamble equivalent to the certain outcome?

and that the relevant questions can be answered meaningfully only by some of the patients. The utility of artificial speech is equal to p (the SG Eq. (2.5.1)). These utilities are substituted in Figure 3.1.1 and the treatment with the highest EU is preferred. For example, if Figure 3.1.2 reveals that U(artificial speech) is 0.9, then EU(radiotherapy) ¼ 0.6  1 þ 0.16  0.9 þ 0.24  0 ¼ 0.744 and EU(surgery) ¼ 0.7  0.9 þ 0.09  0.9 þ 0.21  0 ¼ 0.711. Consequently, radiotherapy has the highest EU and is recommended. This method for applying EU is commonly used in the health domain. With the probability estimates provided by epidemiologists, utilities of patients are elicited using simple SG questions, or alternative methods for measuring quality of life. Advanced computer programs for analyzing complex decision trees have been developed (such as Arborist from Texas Instruments, or TreeAge Pro; see www. treeage.com/resources/academic.html). Such EU analyses are used for individual decision situations but also at the level of policy making and as part of large cost– benefit analyses (Drummond et al. 1987; Gold et al. 1996; Sox et al. 1986; Weinstein et al. 1980). Exercise 3.1.1.b Determine the switching utility b for artificial speech, in Figure 3.1.1. That is, determine the utility of artificial speech that makes the two treatments equivalent. If the utility of artificial speech is higher than the switching utility then surgery is optimal, and if the utility of artificial speech is lower then radiotherapy is optimal. □ 3.2

Risk aversion

Henceforth, outcomes are monetary again. We consider the classical account of insurance in terms of EU theory. We will analyze risk aversion (Definition 2.4.1) under EU in this chapter using Figure 3.2.1. Risk aversion will then amount to concave utility. For other models considered later the implications will be different, and need not imply concave utility. For instance, Example 6.3.1 will give risk aversion with a strictly convex utility function under the rank-dependent generalization of EU. For now, however, we restrict attention to EU where the classical conclusion of concave utility will follow. Note that the x-axis of the figure represents money. For instance, pa þ (1p)b on the x-axis designates a monetary amount, and not a prospect.

Chapter 3: Applications of EU

72

U U(β)

U(pα + (1− p)β)

∗

pU(α) + (1 − p)U(β)

∗

U(α)

α p For the prospect expected value.

1−p

β

pα+(1− p)β α β

, the expected utility, ∗, is lower than

$

∗, the utility of the

Figure 3.2.1 Risk aversion.

It is the expected value of the corresponding prospect. Because of linearity of the diagonal drawn, the height indicated by the small asterisk * at outcome pa þ (1p)b (this outcome is the p/1p mixture of outcomes a and b) is the p/1p mixture of the heights. Hence it is pU(a) þ (1p)U(b), which is the EU of the prospect apb. The properties of utility that correspond to risk aversion, risk neutrality, and risk seeking are (see Figure 3.2.2): concavity : Uðlaþð1lÞbÞ  lUðaÞþð1lÞUðbÞ for all 0 < l < 1 and a;b;ð3:2:1Þ linearity1 : Uðlaþð1lÞbÞ ¼ lUðaÞþð1lÞUðbÞ for all 0 < l < 1 and a;b; ð3:2:2Þ convexity : Uðlaþð1lÞbÞ  lUðaÞþð1lÞUðbÞ for all 0 < l < 1 and a;b: ð3:2:3Þ

U

U

U concave

U

U linear

U convex

Figure 3.2.2 Concavity, linearity, and convexity. 1

A comment for mathematical readers: As commonly done in empirical fields, linear is used in the same sense as affine and does not require U(0) ¼ 0. Because utility is unique up to unit and level, U(0) ¼ 0 can always be obtained.

3.2 Risk aversion

73

Theorem 3.2.1. Under Structural Assumption 3.0.1 (decision under risk and EU): (a) [Risk aversion , U is concave]; (b) [Risk neutrality , U is linear]; (c) [Risk seeking , U is convex]. □ Exercise 3.2.1. (a)b Show that risk aversion holds when restricted to prospects with no more than one or two outcomes if and only if U is concave. (b)c Prove Theorem 3.2.1 in general (for all prospects). □ The result in part (a) of the theorem is central to EU theory. It was first obtained, for two-outcome prospects, by Marshall (1890 Footnote IX in Mathematical Appendix). Concave utility corresponds to diminishing marginal utility, which is a natural assumption: “all these successive units have for their possessor an intensity of utility decreasing from the first unit which responds to the most urgent need to the last, after which satiety sets in” (Walras 1896 p. 98). For a prospect, the risk premium is the difference between the expected value of the prospect and its certainty equivalent. Example 3.2.2. Consider the prospect 80½(20) and assume U(a) ¼ 1  exp(ya) with y ¼ 0.005. U is a concave function. The expectation of the prospect is 30. Straightforward calculations (Exercise 3.2.2) demonstrate that its certainty equivalent is 23.81, which is less than the expectation in agreement with risk aversion. The risk premium is 30  23.81 ¼ 6.19. □ Exercise 3.2.2.a Verify the calculations in Example 3.2.2. □ Exercise 3.2.3.b (a) Indicate on the x-axis of Figure 3.2.1 where the certainty equivalent of the prospect lies. Which distance in the figure corresponds to the risk premium? (b) Show that the risk premium is nonnegative for all prospects if and only if risk aversion holds. □ The risk premium is important for insurance. Assume that during the next year you are bearing a risk – e.g., burglary in your house – that is, a probability distribution over negative outcomes. Assume that the expected value is $100. If you are risk averse, your certainty equivalent will be lower; say, $120. This means that you are willing to pay an insurance premium of up to $120 to insure against burglary. Because the insurance company has many clients, offering such an insurance will cost the insurance company on average $100 per client (by the law of large numbers that some of the readers may know from probability theory courses). Hence, it can make up to $20 by offering you (and people like you) insurance against burglary. The risk premium describes the exact profit margins for insurance, and is the basis of insurance.

74

3.3

Chapter 3: Applications of EU

Applications of risk aversion

Exercise 3.3.1.a Consider bike insurance. Assume that for the relevant amounts of money, which are moderate, your utility is linear. Does EU recommend insuring your bike? □ Exercise 3.3.2.b You play in the final of the TV game show “Deal or No Deal.” There are 26 numbered suitcases. There are also 26 predetermined and known prizes between $0.05 and $5 106. Each suitcase contains one prize, but it is unknown which suitcase contains which prize.2 You randomly selected one of 26 numbered suitcases, namely no. 13. All suitcases not selected by you except one, no. 7, have been opened after that, and only the prizes $0.05 and $2.5  106 are left. One of these two prizes is, therefore, contained in your suitcase no. 13, and the other is in suitcase no. 7. As part of the TV game, a bank that does not know more about the content of the suitcases than you do, offers you $106 for your suitcase. What do you prefer, keeping your suitcase or accepting the bank offer? And what would your hypothetical preference be between suitcase no. 7 and the bank offer? Most people will prefer the sure prize $106 in each case. Show that a Dutch book can be made against these preferences (EV is violated). Show that EU can accommodate such preferences. □ The following exercise concerns an important skill: developing a decision tree to model a decision situation. Exercise 3.3.3c (from Bierman & Fernandez, 1995 Exercises 10.5 and 10.6). An executive at a publishing house has just been given two stock options as a bonus. Each option gives the executive the right (but not the obligation) to purchase one share of the publishing company’s stock for $50, as long as he does it before the closing of the stock market tomorrow afternoon. When the executive exercises either option, he must immediately sell the stock bought from the company at the market prize in effect at the time. An option can only be exercised once. The stock’s price today is $55. The executive knows this price today before deciding what to do today. Hence, if he exercises either option today he is guaranteed a profit of $5 per option exercised. The stock price tomorrow will either be $45 or $65 with equal probability. The executive will know the price tomorrow before deciding what to do that day. This means that if he waits until tomorrow and the stock price rises to $65, he can exercise any remaining options for a profit of $15 per option exercised. On the other hand, if the stock price falls to $45, then exercising either option would result in a loss of $5 per option exercised, which the executive obviously will not do, then preferring to let any option left expire unused. Today the executive can: (1) exercise both options; (2) exercise one option today and wait until tomorrow to decide about the second one; or, (3) exercise neither

2

This program is broadcasted in many countries in the world. The stakes mentioned are used in the Netherlands where the program was invented and first transmitted, and these stakes are the highest in the world.

3.3 Applications of risk aversion

75

option today and wait until tomorrow to decide what to do about both. Tomorrow the executive must either exercise any option(s) not already cashed in or let it (them) expire unused. Assume that there are no brokerage commissions or taxes. Draw a decision tree. Suppose that the executive maximizes EU, with a utility function given by U(0) ¼ 0, U(5) ¼ 10, U(10) ¼ 18, U(15) ¼ 25, U(20) ¼ 31, U(25) ¼ 36, and U(30) ¼ 40. What is the best decision for the executive? You may want to verify that you drew the right decision tree before doing the calculations. □ For prospect (p1:x1,. . .,pn:xn), an elementary mean-preserving spread replaces an outcome xj and its probability pj by two outcomes M, m with probabilities qpj and (1q)pj such that the expectation of the prospect is unaltered. That is, xj ¼ qM þ (1q)m. The resulting prospect is (p1:x1,. . .,pj1:xj1,pjq:M,pj(1q):m,pjþ1:xjþ1. . . pn:xn). Figure 3.3.1 illustrates the case. A mean-preserving spread results from a finite number of replacements of this kind, in other words, from a finite number of elementary mean-preserving spreads.3 The opposite of mean-preserving spread is called mean-preserving contraction. Aversion to mean-preserving spreads means that mean-preserving spreads are less preferred than the original prospect. It can be verified that every prospect is a mean-preserving spread of its expectation (elaboration of Exercise 3.2.1). Hence, aversion to mean-preserving spreads implies risk aversion. For general decision models, aversion to mean-preserving spreads, sometimes called strong risk aversion, is a more restrictive condition than risk aversion (sometimes called weak risk aversion), as we will see later. Under EU, however, and

p1 x1 . . .

p1 . . .

pj

pj

. . . pn

xj

xn

x1

q

. . . pn

1−q

M

m

xn

qM + (1−q)m = xj, so that the means are the same.

Figure 3.3.1 Aversion to elementary mean-preserving spreads.

3

For prospects with infinitely many outcomes, every outcome is replaced by a conditional distribution with the same expectation, which may require infinitely many elementary mean-preserving spreads and limit-taking. This book considers only finitely many outcomes.

76

Chapter 3: Applications of EU

its implication of independence, it is not difficult to infer from Figure 3.3.1 that risk aversion implies aversion to mean-preserving spreads (elaboration of Exercise 3.2.1). Observation 3.3.1. Under EU, aversion to mean-preserving spreads is equivalent to risk aversion, which is equivalent to concavity of utility. □ A small variation of aversion to mean-preserving spreads is known as second-order stochastic dominance; it adds first-order stochastic dominance. Alternative equivalent definitions in terms of integrals have often been used (Bawa 1982). These conditions can be useful if we do not know the utility function precisely or have incomplete preferences. For example, if all we know about utility is that it is concave, then already we can conclude what the preference must be between a prospect and its meanpreserving spread. If we have incomplete preferences, and only prefer a prospect to another if it has higher EU for all concave utility functions, then we can still conclude that a prospect is preferred to any mean-preserving spread. See Bawa (1982) and Levy (1992) for early surveys. Baucells & Heukamp (2006) provided a recent application of such techniques to prospect theory. Using such techniques, Baucells & Shapley (2008) and Dubra, Maccheroni, & Ok (2004) independently and simultaneously found an appealing extension of Theorem 2.6.3 to incomplete preferences. Such extensions are important given that completeness is the most questionable condition in Theorem 2.6.3. Assignment 3.3.4.a Assume that U(a) ¼ 1  exp(a/100). What is the risk premium of 100½0? □ Assignment 3.3.5 [Risk aversion versus variance].c This assignment demonstrates that risk aversion need not always imply aversion to increased variance. Assume risk aversion (concave utility). Throughout, x and y are two prospects with the same expectation, where x has a larger variance. (a) Assume x ¼ x1½x2 and y ¼ y1½y2 with x1 > x2 and y1 > y2. Show that y ≽ x. Hence, risk aversion does imply aversion to increased variance for two-outcome prospects. (b) Consider three-outcome prospects x ¼ ⅓x1,⅓x2,⅓x3 and y ¼ ⅓y1,⅓y2,⅓y3. Give an example where U and x, y are such that x  y. That is, higher variance may be preferred despite risk aversion. □ Assignment 3.3.6.c This assignment demonstrates that risk aversion implies arbitrage. More generally, it shows that every deviation from risk neutrality implies arbitrage. You may assume that risk neutrality for all 50–50 prospects implies complete risk neutrality.4 Show that arbitrage is possible whenever there is no risk neutrality, for instance as soon as there is strict risk aversion. A difficulty in this assignment is that we have defined risk neutrality for decision under risk, and arbitrage has been 4

This follows mainly from Theorem 1 of }2.1.4 of Acze´l (1966), or from Hardy, Littlewood & Po`lya (1934 Theorems 111 and 86). Note that their term convex has the same meaning as the modern term midpoint-convex. The result follows first on the interior, then by continuity also for the boundary. This shows for instance that midpoint concavity implies concavity under continuity or monotonicity.

3.4 Indexes of risk aversion

77

defined for decision under uncertainty. You, therefore, have to understand }2.1–2.3 to be able to do this assignment. □ 3.4

Indexes of risk aversion

Consider two persons with preference relations ≽1 and ≽2, respectively, over prospects. Assume that they maximize EU with utilities U1 and U2. We say that ≽2 is more risk averse than ≽1 if [a 1 x ) a ≽2 x] for all prospects x and outcomes a. In other words, whenever a riskless prospect a is equivalent to a risky prospect x for person 1, then person 2, being more risk averse, prefers the riskless prospect. Theorem 3.4.1. Person 2 is more risk averse than person 1 if and only if U2(·) ¼ j(U1(·)) for a concave transformation j. □ Theorem 3.4.1 is an almost trivial corollary of Theorem 3.2.1a, taking as unit of outcome not money, but utility of person 1, as will be elaborated in }3.9. If utility U is twice continuously differentiable then a well-known measure of concavity of utility is U00 /U0 , the Pratt–Arrow measure. Because concavity of U corresponds with risk aversion under EU, the measure can also be taken as an index of risk aversion here. Whereas risk aversion cannot be defined for general outcomes if expected value and the corresponding risk neutrality level cannot be defined, comparative risk aversion then still is meaningful (Yaari 1969). Theorem 3.4.1 holds with no alteration for general outcomes (Peters & Wakker 1987 Theorem 2). These observations suggest that comparative risk aversion is the more basic concept, with absolute risk aversion resulting from a choice of a risk neutrality level. Exercise 3.4.1.!c Assume that U1 and U2 are twice continuously differentiable with positive first derivative everywhere. Show that ≽2 is more risk averse than ≽1 if and only if U200 /U20   U100 /U10 everywhere. □ Exercise 3.4.1 illustrates that the Pratt–Arrow measure is indeed a useful index of risk aversion. Hence, it also is a useful index of concavity of utility. Sometimes the term risk tolerance is used to designate the reciprocal U0 /U00 of the Pratt–Arrow measure. The best interpretation of these concepts will be given later in Exercise 3.5.5b, where an empirical meaning will be given to them. On the domain of positive outcomes a > 0, an alternative popular measure of risk aversion, or concavity of utility, is a  (U00 (a)/U0 (a)). It is sometimes called the relative (or proportional) measure of risk aversion. It is the most used measure of risk aversion in the expected utility literature today. The Pratt–Arrow measure is also called the absolute measure of risk aversion. One of the main messages of this book is the following observation: Under models more general than EU, risk aversion is no longer equivalent to concave utility. It then depends on other factors besides utility. We will study those factors in later chapters. The Pratt–Arrow measure of utility then no longer is an index of risk aversion,

Chapter 3: Applications of EU

78

but a measure of concavity of utility. Unfortunately, this observation concerns a widespread misunderstanding in the economic literature today. Although economists often assume that EU is violated descriptively, and assume more general models, they nevertheless continue to use the logical implications that they have learned only under EU and that need not hold for the more general models. They thus continue to use the term risk aversion to refer to concave utility, and indexes of concavity of utility are still called indexes of risk aversion, even though this terminology is clearly inappropriate when EU is abandoned. It is preferable in more general contexts to simply call concavity of utility what it is (concavity of utility) and not what it is not (risk aversion). Remember, therefore, that concavity of utility and the Pratt–Arrow index can be related to risk aversion only if EU is assumed. The latter assumption is made in this chapter. An interesting case occurs when U1 is the utility U of a person and U2 the utility of that same person when richer by an amount ε > 0, that is, we compare U2(a) ¼ U(a þ ε) with U(a). If for all ε > 0, U2 is less concave than U1, then we say (assuming EU as throughout this chapter) that the person exhibits decreasing risk aversion. This holds if and only if  U00 /U0 is decreasing. Increasing risk aversion refers to U2 being more concave than U1 for all ε > 0. 3.5

Parametric families of utility

Wakker (2008a) provides a more extensive didactic account of the material presented in this section. Power utility. The most popular parametric family of utility is the power family, in economics often called the family of constant relative risk aversion (CRRA). With the parameter denoted y, we define the family on Ñþ . For a > 0: for y > 0; UðaÞ ¼ ay ; for y ¼ 0; UðaÞ ¼ lnðaÞ; for y < 0; UðaÞ ¼ ay :

ð3:5:1Þ

Multiplication by 1 for negative y serves to have the function increasing. For y > 0 the function can also be defined at a ¼ 0, where it then is 0. For y  0 the function is 1 at a ¼ 0. For y ¼ 1 the function is linear, corresponding to expected value and risk neutrality. As Figure 3.5.1 will illustrate, y is an anti-index of concavity. The smaller y is, the more concave U is and the more risk aversion will be generated. In economics, the parameter 1y, called the (relative or proportional) index of risk aversion, is often used to describe the family. Under EU, the functions U are interval scales and we can replace them by any function t þ sU for t 2 Ñ and s > 0. We can thus define concisely: For all y 6¼ 0; UðaÞ ¼ ay=y;

ð3:5:2Þ 0

which amounts to the scaling U (1) ¼ 1.

3.5 Parametric families of utility

79

q=1 q=0 (= In)

Θ

=

–5

Θ=

–2

0

1

Θ

=

−2

=2

Θ

=3 0

Θ

=5

Θ

0

1

2

Θ Θ = = 0. −0. 1 Θ= 1 −0 Θ = .5 −1

Θ=

0.5

–1

Figure 3.5.1 Power utility curves, normalized at 1 and 2.

The definition of the power family may seem to be ad hoc concerning the transition from y > 0 to y < 0, but Wakker (2008a) demonstrated that this transition is natural, and in fact is the only natural way to define the family. Figure 3.5.1 depicts some power functions normalized at a ¼ 1 and a ¼ 2. That is, they were obtained from a power function U using the normalization UðaÞ  Uð1Þ : Uð2Þ  Uð1Þ Figure 3.5.1 illustrates that the transition from y > 0 to y < 0 is natural. The following property, imposing invariance of preference under a change of unit of the outcomes, will characterize power utility for positive outcomes.

80

Chapter 3: Applications of EU

The preference between two prospects is not affected if all outcomes of both prospects are multiplied by the same positive number l > 0: ð3:5:3Þ This condition is usually called constant relative risk aversion, whence power utility is often called the CRRA family in the economic literature. Then the index of relative risk aversion, defined as aU00 /U0 , and equal to 1y and independent of a, is commonly used as an index of risk aversion as explained in }3.4.1. We avoid this terminology because under the nonexpected utility models studied later in this book, risk aversion and risk attitudes can no longer be equated with utility curvature, but depend also on other factors. Better terminologies would result with concavity of utility instead of risk aversion, but we will not pursue this point, and will leave the condition as a numbered equation. Observation 3.5.1. Assume EU with continuous strictly increasing utility, restricted to positive outcomes. Then utility is from the power family if and only if Eq. (3.5.3) (invariance of preference under a change of unit) holds. □ Exercise 3.5.1.a Show that the condition in Eq. (3.5.3) is satisfied for power utility. □ Assuming power utility, a choice between 100 for sure or 200½50 can serve as a diagnostic for observing the sign of the power. For power 0 (logarithmic utility) indifference results, with the risky prospect either yielding ln(2) utility units more or less than the sure prospect. There is more risk aversion and a negative power if the sure 100 is preferred and more risk seeking with a positive power if the risky prospect is preferred. Exponential utility. Another popular family of utility is the exponential family, often called the family of constant absolute risk aversion (CARA) in the economic literature, and defined on the entire outcome set Ñ: for y > 0; UðaÞ ¼ 1  eya ; for y ¼ 0; UðaÞ ¼ a; for y < 0; UðaÞ ¼ eya  1:

ð3:5:4Þ

It is the power family applied to ea rather than to a, and with y instead of y. Hence, the transition from y > 0 to y < 0 is, again, not ad hoc but natural. The parameter y is an index of concavity, with linear utility for y ¼ 0, concave utility for y > 0, and convex utility for y < 0. Again, the functions U are interval scales under EU and we can replace them by any function t þ sU for t 2 Ñ and s > 0. We can thus define concisely: For all y 6¼ 0; UðaÞ ¼ ð1  eya Þ=y;

ð3:5:5Þ

which amounts to the scaling U0 (0) ¼ 1. This form, behaviorally equivalent to the form in Eq. (3.5.4), has more convenient numerical and analytical properties, making it useful for data fitting.

3.5 Parametric families of utility 2

θ = −2

U

1

−2

81

θ=2

−1 2

1 θ = −0.6

$ −1

θ=0 θ = 0.6

−2

Figure 3.5.2 Exponential utility, normalized at 0 and 1.

The following property, imposing invariance of preference under a change of level of the outcomes, will characterize exponential utility: The preference between two prospects is not affected if all outcomes of both prospects are increased by the same term m:

ð3:5:6Þ

This condition is usually called constant absolute risk aversion, whence exponential utility is often called the CARA family in the economic literature. Similar objections apply to this terminology as to the CRRA terminology discussed before. Exercise 3.5.2.!a Assume exponential utility. Show that the index of absolute risk aversion, U00 /U0 , is y. It is independent of a. □ Observation 3.5.2. Assume EU with continuous strictly increasing utility. Then utility is from the exponential family if and only if Eq. (3.5.6) (invariance of preference under a change of level of the outcomes) holds. □ Exercise 3.5.3.a Show that Eq. (3.5.6) is satisfied for exponential utility. □ Exercise 3.5.4.b This exercise illustrates Eqs. (3.5.3) and (3.5.6). Consider the prospects S (“safe”) ¼ 30 and R (“risky”) ¼ 100½0. The criterion for being “close to” will be a distance less than 0.1. We write CE1 for the certainty equivalent of 200½0 and CE2 for the certainty equivalent of 200½100. (a) Assume that U(a) ¼ ay with y > 0. Determine the preference between the prospects for all 0 < j  100 and y ¼ j/100, and find an analytic expression for the y that yields indifference between S and R. Give a three-digits approximation of the indifference-y. Assuming this value of y, give an analytic

Chapter 3: Applications of EU

82

expression for CE1 and CE2, and give a three-digits approximation for each. Is CE1 close to twice 30, or is it considerably larger or smaller? Is CE2 close to 30 þ 100, or is it considerably larger or smaller? (b) Assume that U(a) ¼ 1  eya with y > 0. You can spend some time trying to find an analytic solution for y under indifference between S and R, but will not succeed. Let your computer determine the preference between the prospects for all 0 < j  100 and y ¼ 0.001 þ j/1000. Give a three-digits approximation of the indifference-y. Assuming this value of y, give an analytic expression for CE1 and CE2, and give a three-digits approximation for each. Is CE1 close to twice 30, or is it considerably larger or smaller? Is CE2 close to 30 þ 100, or is it considerably larger or smaller? □

Empirical meaning of risk tolerance Exercise 3.5.5.c Assume exponential utility with y > 0, and consider the following prospects, for various b. ½

½

β

–β/2

(a) Express the optimal value of b numerically in terms of y. (b) Show that the prospect is indifferent to not gambling (receiving $0 with probability 1) for b  0.96/y  1/y. □ Part (b) gives an easy way to measure the Pratt–Arrow index y and its reciprocal, risk tolerance, empirically (assuming EU!): Observation 3.5.3. If your risk tolerance is constant, then it is approximately the maximum stake for which you are willing to take a 50–50 gamble if in case of a loss, half of the loss gets reimbursed. □ At the level of your risk tolerance, losses seem to weigh about twice as much as gains, with a gain 1=y offsetting a loss 1=2y. Later, when we introduce prospect theory, we will consider other interpretations of such factors (“loss aversion”). A convenient aspect of risk tolerance is that its unit is money, that is, your risk tolerance is simply a monetary amount. This makes it easy to conceptualize and to avoid confusion about its unit. The results of this section also hold for general intervals as domain. This follows from Lemma 1 in Miyamoto (1983), applied to U(ln(a)) or U(a), respectively. Observation 3.5.4. Observation 3.5.1 also holds if the domain of utility is an arbitrary subinterval of Ñþ . Observation 3.5.2 also holds if the domain of utility is an arbitrary subinterval of Ñ: □

3.6 Data fitting for EU

83

Empirical findings. The common finding of empirical measurements of utility under EU is: (a) Absolute risk aversion, measured by the Pratt–Arrow measure U00 (a)/U0 (a), decreases in wealth; (b) relative risk aversion, measured by the index aU00 (a)/U0 (a) (only for a > 0), increases in wealth.5 A richer person will then choose larger absolute, but smaller proportional, risky investments. Utility should then be between exponential and power. It led Abdellaoui, Barrios, & Wakker (2007) to introduce a new parametric family of utility, called expo–power utility, which is between exponential and power utility. The preceding empirical findings on utility fit well with the ratio–difference principle from psychology (Stevens & Davis 1938; Baron 1997). Here subjects, when carrying out quantitative tasks, as a heuristic either carry out additive operations or multiplicative operations. Then group averages give results between constant additive and constant multiplicative attitudes. Constant relative risk aversion was found by several studies.6 Uncommon findings exist.7 3.6

Data fitting for expected utility under risk

This section gives numerical examples and exercises to illustrate how EU can be used to describe and predict preferences using data fitting techniques. We fit the data by minimizing the distance between predictions of a theory and observed data described in }A.2, and based on calculations by computer. We usually assume parametric families of utility. Exercise 3.6.1a [Deterministic fit (no error and perfect fit)]. We observe the following indifferences: 0.01  10.100, 0.36  10.600, and 0.81  10.900. (a) Show that EU can accommodate these observations. Set U(0) ¼ 0 and U(1) ¼ 1 and determine the utility of the outcomes 0.01, 0.36, and 0.81. (b) Show that CE(0.812/30) ¼ 0.36. (c) Assume that utility is a power function, that is, U(a) ¼ ay for some y > 0. What is y? Give the predicted CE’s of 10.800 and 0.800.700.25, and the predicted preference between them. □ Exercise 3.6.2b [Measurement with errors and fitting]. Assume that we observe the three indifferences of Exercise 3.6.1, but in addition we observe a fourth indifference, 0.81⅔0  0.40. As part (b) of Exercise 3.6.1 showed, EU must be violated. Find the power y ¼ j/100 > 0 (for an integer j) such that EU with power utility U(a) ¼ ay best fits the data in the sense of }A.2. Give the distance. Predict the CE’s of 1⅔0 5

6 7

Arrow (1971 p. 96); Binswanger (1981); Campbell (1996); Fetherstonhaugh et al. (1997); Holt & Laury (2002); Kachelmeier & Shehata (1992); Meyer & Meyer (2005); Ogaki & Zhang (2001); Post et al. (2008); Yesuf & Bluffstone (2009). Brunnermeier & Nagel (2008); Friend & Blume (1975); Harrison, Lau, & Rutstro¨m (2007); Pa¨lsson (1996); Rapoport (1984); Szpiro (1986). Blake (1996); Cohn et al. (1975 p. 606); Gollier (2001b p. 187) found decreasing RRA. Cohen & Einav (2007) found increasing ARA.

84

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and 0.81⅔0.20, and the preference between these prospects. If you do not know how to proceed, read the footnote.8 □ Exercise 3.6.3.c Consider the following seven choice pairs, where preferences have to be substituted for question marks: x1 ¼ 3000.500 ? 20, x2 ¼ 2000.500 ? 100, x3 ¼ 2000.010 ? 10, x4 ¼ 1000.050 ? 14, x5 ¼ 960.500 ? 39, x6 ¼ 720.950 ? 55, x7 ¼ 1000.950 ? 78. As an index of risk aversion we take the number of safe choices made. Assume that the median number of safe choices in our data set is 3. There is no analytic solution to the following problems. Obtain the results by trial and error, using your computer to calculate the result for many y. (a) Translate the finding of 3 safe choices into the relative risk aversion index often used in the economic literature. That is, assume EU with a power utility U(a) ¼ ay and determine the powers y that imply 3 safe choices. (The index of relative risk aversion is 1y.) Assume for simplicity that y > 0. You may assume, and your calculations will verify it, that the number of safe choices decreases in y. (b) Translate the finding of 3 safe choices into the absolute Pratt–Arrow risk aversion index. That is, assume EU with exponential utility U(a) ¼ 1exp(ya) and determine the values y (the Pratt–Arrow index of risk aversion) that imply 3 safe choices. If you do not know how to proceed, suggestions are given in the footnote.9 □ Exercise 3.6.4.c Consider the following certainty equivalent measurements for seven risky prospects xj. x1 ¼ 3000.500  112, x2 ¼ 2000.500  45, x3 ¼ 2000.010  6, x4 ¼ 1000.050  8, x5 ¼ 960.500  23, x6 ¼ 720.950  34, and x7 ¼ 1000.950  45. Under EU with utility to be specified, determine which model best fits the data, using the distance measure of }A.2. (a) Assume exponential utility, that is, U(a) ¼ 1exp(ya) for all y > 0. Determine the optimal value of y (¼ the Pratt–Arrow index of risk aversion) for all y ¼ j/1000 for 0 < j  100. Report the optimal y and the corresponding distance, and the distances for y  0.001, y  0.002, y þ 0.001, and y þ 0.002. If you do not know how to proceed, read the footnote.10 Predict the CE’s of 300⅔250 and 285⅔276, and the preference between these prospects. 8 9

10

For a y considered, first calculate EU of the prospects, then the predicted certainty equivalents U1(EU) (¼ EU1/y for y > 0); and so on. First verify that y ¼ 0 has only two safe choices and one indifference, so that it just fails to fit the data. Then verify that y ¼ 0.00001 generates too much risk seeking because there are only two safe choices. You may assume that all y < 0 similarly generate too much risk seeking. Then calculate the EU differences for y ¼ 0, 0.1, 0.01, 0.001, 0.005, 0.008, and 0.009. On the basis of these calculations, find (up to a difference of 0.001) the largest y > 0 such that for larger y there is too much risk aversion and there are more than three safe choices, but for y between this upper bound and y ¼ 0 there are exactly three safe choices. You may assume, and your calculations will verify it, that the number of safe choices increases in y. For the y considered, first calculate the EU of the prospects, then the predicted certainty equivalents U1(EU) (¼ ln(1EU)/y for y > 0), and then the squared distances; and so on.

3.7 Multiattribute utility

85

(b) Assume power utility, U(a) ¼ ay. Determine the optimal value of y over all y ¼ j/ 100 for j  10. Predict the CE’s of 300⅔250 and 285⅔276, and the preference between these prospects. (c) (This part is best answered after parts (a) and (b) have been done and their answers have been verified.) Which family fits better, the exponential family or the power family? Which preference between 300⅔250 and 285⅔276 would you recommend? Additional Comment [Motivating distance measure chosen]: The distance measure refers to differences between certainty equivalents, designating amounts of money, for both parametric families. It, therefore, makes sense to compare them. Had we taken EU differences as distance measure, then these would not have been related very clearly to objects with a physical meaning, and they would have been more difficult to compare. For example, halving one utility function does not change any empirical implication, but would affect the EU distances and the comparisons between the different utility functions. □ Exercise 3.6.5.b The following eight indifferences are median observations from Tversky & Kahneman (1992): gains: 1500.0550  64; 1000.500  36; 1000.950.00  78; losses: 00.75(100)  23.50; 00.50(50)  21.00; 00.05(100)  84; mixed: 1010.50(50)  0; 500.50(20)  1120.50(50). We use these eight indifferences to assess the utility function of EU. We then use the assessed EU model to make a prediction. To avoid some analytical inconveniences of power utility at negative inputs and 0, we assume exponential utility: U(a) ¼ 1  exp(ya). Find the 0.0000 < y ¼ j  0.0001 < 0.0100 that best fits the data. Give the distance according to }A.2. Predict CE(1000.05(200)). □ Exercise 3.6.50 .b The median value of CE(1000.05(200)) found by Tversky & Kahneman (1992) was 179. Determine by how far your prediction in Exercise 3.6.5 is off from the actually observed value. □ Assignment 3.6.6.a We observe the following indifferences: 10  1200.100, 30  600.600, 81  1000.900. Assume power utility, with U(a) ¼ ay for 0 < y ¼ j/100 < 2. Which y fits best? Predict the CE’s of 1000.800 and 800.8025, and the preference between these prospects. □

3.7

Multiattribute utility

This section assumes, in deviation from most of this book, that outcomes need not be monetary. Example 3.7.1. Consider a decision about a medical treatment where it is uncertain what the resulting health state will be, but it is understood that the health state

86

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can then be expected to be stable for the rest of one’s life. (Q,T) refers to the chronic health state of living T years in health state Q and then dying. The set of outcomes is the product of a set of health states and a set of possible life durations where the latter contains 0 (and no negative numbers). We assume EU with U(Q,T) the utility of chronic health state (Q,T). We seek to simplify the task of determining U for each (Q,T), and formulate some plausible preference conditions. For simplicity, we assume monotonicity in life duration, i.e. (Q,T þ ε)  (Q,T) for all ε > 0. This rules out negative and null health states such as death. The zero-condition requires that all (Q,0) are equivalent, which is a self-evident requirement. Assume further that we consider it reasonable that SG equivalences over life duration be independent of the health state assumed (Figure 3.7.1), a condition that we only need when one prospect outcome is immediate death. □ p (Q,T) ∼

1−p

(Q,M) (Q,0)

⇒ (H,T) ∼

p

(H,M)

1−p

(H,0)

Figure 3.7.1 SG invariance.

Observation 3.7.2. The following two statements are equivalent. (i) Monotonicity in life duration, SG invariance, and the zero-condition hold; (ii) U is multiplicative, i.e. UðQ; TÞ ¼ VðQÞ  WðTÞ

ð3:7:1Þ

for some functions V and W with W(0) ¼ 0, W strictly increasing in T, and V(Q) > 0 for all Q. □ Exercise 3.7.1.c Prove Observation 3.7.2. This result was proved by Miyamoto et al. (1998 Theorem 3.1). □ V designates quality of life and W designates the value of life duration. W incorporates discounting. The model is the generalized QALY (quality of life) model. The QALY model concerns the special case of a linear W. Eq. (3.7.1) greatly simplifies our task. We can separately determine quality of life (V) and the value of life years (W), and need not consider all combinations of quality of life and life duration separately. This example illustrates once more how behavioral foundations help to find the right model, identifying the parameters and concepts relevant for our decisions. As virtually all preference conditions in this book, SG invariance requires consistency between different measurements. □ In general multiattribute utility, the outcome set, denoted X, is a product set X1   Xm. Xi may be a general set. Outcomes are denoted a ¼ (a1,. . .,am), where ai is the ith attribute (or coordinate) of a. The set Xi is the ith attribute set (or component). Here superscripts designate attributes and not powers. I use superscripts because subscripts refer to the different outcomes of a prospect. In Example 3.7.1, the first attribute concerns quality of life and the second concerns life duration. Another

3.7 Multiattribute utility

87

example results if outcomes designate cars, with m ¼ 3, and with attribute 1 describing the maximum speed, attribute 2 describing the color, and attribute 3 describing the price, under the assumption that these are considered the only relevant properties of cars. In applications, one should incorporate enough attributes to capture the essence of the decision problem, but so few that the model remains tractable. Similar points hold not only for the number of attributes to incorporate in multiattribute utility, but for modeling in general and, for instance, also for the number of states of nature to incorporate in the state space. Especially the art of keeping the model simple by leaving out many details but without losing the essence is crucial (von Winterfeldt 2007). Lancaster (1966) argued that objective attributes (such as speed of a car) are often better regrouped into more fundamental attributes (such as safety of a car). In what follows, we assume that the description of outcomes is complete: all relevant aspects of the car concern the aforementioned three aspects (attributes). More examples are in Appendix D. This section considers the special case of multiattribute utility where: (a) Probabilitycontingent prospects – i.e. probability distributions – are given over the m-tuples; (b) these prospects are evaluated by EU, as assumed throughout this chapter. EU implies a linearity in probability that simplifies the analysis. This section briefly describes some of the main behavioral foundations. A detailed analysis is in Keeney & Raiffa (1976). Fishburn (1982) contains many advanced technical results. Recent applications include Andre´ (2009) and Yang (2008). A jth outcome xj in a prospect x ¼ (p1: x1,. . .,pn:xn) is xj ¼ (xj1,. . .,xjm), with xji the ith attribute of outcome xj; see Figure 3.7.2. As throughout this book, xj denotes the jth outcome of prospect x and a denotes a general outcome that is not related to a particular prospect. p1 p2

pn

(x11,..., x1m) m 1 . (x2 ,..., x2 ) → . .

p1U (x11,..., x1m) + ... + pnU (xn1,..., xnm)

(xn1,..., xnm)

Figure 3.7.2 A prospect with multiattribute outcomes and its expected utility.

We examine how different attributes are aggregated into an overall evaluation. That is, we examine how an evaluation U(a1,. . .,am) of an outcome can be obtained as an aggregation of attribute utility functions Ui(ai). Aggregation questions are central to many domains. Decision under uncertainty, the main topic of this book, can be considered a special case of aggregation, with states of nature as attributes. This will be central to }4.9.3. A prospect p1x1. . .pnxn over X generates a marginal prospect p1x1i. . .pnxni over the attribute set Xi. Figure 3.7.3 depicts an example that will now be explained. Consider, for m ¼ 2, the prospect ð5years, blindÞ1/ 2 ð20years, healthyÞ:

ð3:7:2Þ

Chapter 3: Applications of EU

88

½

½

(5 years, blind)

Marginal for life duration

(20 years, healthy)

½

½

(5 years) (20 years)

Prospect of Eq. (3.7.2) ½ ½

½

(5 years, healthy)

Marginal for health

½

(blind) (healthy)

(20 years, blind)

Prospect of Eq. (3.7.3)

The marginals

Figure 3.7.3 Two prospects with the same marginals.

Here outcomes are chronic health states (defined in Example 3.7.1). The first attribute of the chronic health state designates the duration and the second designates the health state. The prospect in Eq. (3.7.2) generates the marginal distribution 5years½20years over life duration and the marginal distribution blind½healthy over health states. Marginal distributions lose information about joint distributions, including correlations. For example, ð5years, healthyÞ1/ 2 ð20years, blindÞ;

ð3:7:3Þ

generates the same marginals as the prospect in Eq. (3.7.2). Multiattribute risk aversion and additive decomposability. For attribute gi ∈ Xi and outcome a, the outcome gia denotes outcome a with attribute ai replaced by gi. Assume that you own a prospect gia½gib. You then know for sure that the resulting ith attribute value will be gi. Assume that gia ≼ gib. Imagine that you can choose to have gi improved into a preferred attribute level di for either gia or gib in the prospect. That is, you can choose between dia½gib and gia½dib. Multiattribute risk aversion holds if you always prefer to have the attribute improved for the worst outcome; i.e., dia½gib ≽ gia½dib. Multiattribute risk seeking holds if the reversed preference always holds (de Finetti 1932; Payne, Laughhunn, & Crum 1984; Pliskin, Shepard, & Weinstein 1980 p. 210; Richard 1975). Before reading the following text, you are invited to determine your preference between the chronic health states in Eqs. (3.7.2) and (3.7.3). For chronic health states multiattribute risk seeking is the common finding: Taking prospect (5years, blind)½(20years, blind) as point of departure, people usually prefer the improvement in Eq. (3.7.2), with vision regained for the 20-year period, to the improvement in Eq. (3.7.3), with vision regained for the 5-year period. The intersection of multiattribute risk aversion and multiattribute risk seeking may be called multiattribute risk neutrality. It means that an improvement of an attribute is evaluated the same way independently of whatever the other attribute levels are, and it signals an absence of

3.7 Multiattribute utility

89

interactions between attributes. It can indeed be shown that multiattribute risk neutrality holds if and only if we have an additive decomposition: Uða1 ; . . . ; am Þ ¼ U1 ða1 Þ þ þ Um ðam Þ

ð3:7:4Þ

(Wakker, Stiggelbout, & Jansen 2004 Corollary 3.4(ii)). For such a decomposition there are, indeed, no interactions between the attributes. Fishburn (1965) proposed a stronger condition to characterize additive decomposability, which constitutes a classical result in multiattribute utility theory: Theorem 3.7.3. The additive decomposition of Eq. (3.7.4) holds if and only if the preference value of a prospect depends only on the marginal prospects generated over the attributes (“marginal independence”). □ This result was alluded to by Arrow (1951a pp. 431–432). The condition in Theorem 3.7.3 implies that the prospects in Eqs. (3.7.2) and (3.7.3) are equivalent. Hence, it implies multiattribute risk neutrality. In the presence of EU as assumed here, it is equivalent to multiattribute risk neutrality. Utility independence. Keeney & Raiffa (1976) considered many utility independence conditions, where some attributes of outcomes are kept fixed at a deterministic level, and then prospects over the remaining attributes are considered. Strong utility independence means the following: Imagine that for a subset I of {1,. . .,m}, we fix the attribute level of every attribute in I. Then the preferences over the remaining prospects should be independent of the particular levels at which we fixed the attributes from I. This condition should hold for every subset I. SG invariance in Example 3.7.1 is implied by strong utility independence. We next consider another example. Example 3.7.4. Assume that m ¼ 3, X ¼ Ñ3 . We take I ¼ {2,3}, fix the first attribute at the deterministic level 0, and observe (0,6,7)½(0,2,3)  (0,5,6)½(0,3,4). Then, when fixing the first attribute at the deterministic level 1, strong utility independence implies (1,6,7)½(1,2,3)  (1,5,6)½(1,3,4). □ Theorem 3.7.5. Strong utility independence holds if and only if either U is additively decomposable as in Eq. (3.7.4), or U is multiplicatively decomposable: Uða1 ; . . . ; am Þ ¼ U1 ða1 Þ   Um ðam Þ

ð3:7:5Þ

where no U takes the value 0 or changes sign. □ A weak version of utility independence concerns the condition when only imposed on all sets I containing all but one of the attributes. This allows for multilinear representations with many kinds of interactions between the attributes. Torrance and his collaborators used this general model extensively in the health domain (Furlong et al. 2001; Torrance et al. 1995). Guerrero & Herrero (2005) presented an appealing intermediate result. Attributes 1,. . .,n refer to time points and each set {j,. . .,n} of time points is utility independent from the past. A “semi-separable” utility i

90

Chapter 3: Applications of EU

Uðx1 ; . . . ; xn Þ ¼

 Xn  Yj1 u ðx Þ c ðx Þ j j i i j¼1 i¼1

results, as proved simply using their Eq. 1. Homogeneous attribute sets (Anscombe & Aumann 1963; Harsanyi 1955). In the rest of this section we consider the special case where X1 ¼ · · · ¼ Xm. Then we can equate these sets with a set C ¼ X1 ¼ · · · ¼ Xm, called the set of prizes. Anscombe & Aumann (1963) considered this case, and added a monotonicity condition. I will now explain their condition in the terminology of this section.11 Assume, similarly as in the utility independence conditions of Keeney & Raiffa (1976), that all but the ith attribute are kept fixed at a deterministic level, and prospects over the remaining attribute i are considered. It results in a preference relation over prospects over C that, in general, will depend on i and on the fixed deterministic levels of the other attributes. Under the additive decomposition U1(a1) þ · · · þ Um(am) of Eq. (3.7.4), the resulting preferences then maximize EU with Ui as utility function. A&A monotonicity requires that all resulting preference relations over prospects over C that are not degenerate (one equivalence class; e.g., in Eq. (3.7.4) if Ui is constant) are the same; see their Assumption 1 (“monotonicity in the prizes”). So, they are not only independent of the levels at which the other attributes are kept fixed (as implied by Eq. (3.7.4)), but also of the particular attribute i chosen. This condition together with Fishburn’s (1965) marginal independence condition (Theorem 3.7.3) characterizes a representation Uða1 ; . . . ; am Þ ¼ q1 uða1 Þ þ þ qm uðam Þ

ð3:7:6Þ

for a function u and nonnegative qj that sum to 1. The proof is so simple that we give it here in the main text. Theorem 3.7.6 [Anscombe & Aumann]. Assume X1 ¼ · · · ¼ Xm ¼ C. Then the additive decomposition of Eq. (3.7.6) holds if and only if the preference value of a prospect depends only on the marginal prospects generated over the attributes and A&A monotonicity holds. Proof. Necessity of the conditions follows from substitution. Hence we assume the preference conditions and discuss their implications. By Theorem 3.7.3 we obtain the representation U(a1,. . .,am) ¼ U1(a1) þ · · · þ Um(am) (Eq. (3.7.4)). By A&A monotonicity and the uniqueness result in Observation 2.6.30 (EU for risk), all nonconstant Ui are the same up to level and unit. We can give them all the same level so that they are proportional, and by normalization can scale them as in Eq. (3.7.6). □ Anscombe & Aumann interpreted (a1,. . .,am) as a gamble on a horse race, where m horses participate, exactly one will win, and ai is the prize won if horse i wins. That is,

11

An equivalent account from a different perspective will be given in a footnote in Example 4.9.7.

3.8 Taxation/Welfare

91

they interpreted multiattribute outcomes as event-contingent prospects, and qi as the (subjective) probability of horse i winning the race. As overall representation of prospects we get  Xm     Xn  i  i p1 : x1 1 ; . . . ; x1 m ; . . . ; pn: xn 1 ; . . . ; xn m 7! p q u xj : ð3:7:7Þ j j¼1 i¼1 The representation in Eqs. (3.7.6) and (3.7.7) were also characterized by Harsanyi (1955), who interpreted (a1,. . .,am) as a welfare allocation of good ai for individual i in a society, i ¼ 1, . . ., m. That is, attributes reflect individuals. Harsanyi, in fact, considered more general outcome sets X that need not be full product sets X1  · · ·  Xm, but can be almost any subset. Hence, he did not need to specify the decomposition of X. Instead, he took a general set X of social states, where for social state x we can interpret ai to designate what this state means for individual i. Harsanyi needed only minimal richness assumptions in his proof, such as the requirement that for each individual i a social state is available yielding utility 1 for i and utility 0 for all other individuals. 3.8

Taxation and welfare theory with cardinal utility

Another, controversial, application of EU concerns the allocation of resources over different individuals, such as in welfare theory and taxation. The most natural approach is to maximize the average or, equivalently, the total utility of the group, that is, the sum of individual utilities (utilitarianism). This approach is, however, not well defined if utility is ordinal (determined only up to a strictly increasing transformation). At the beginning of this century, economists came to assume that utility was ordinal and, therefore, refrained from utilitarianism. A famous result is the voting paradox of Arrow (1951b), which attempted to derive group behavior in a strictly ordinal spirit and then ran into a paradox. Some authors have argued, and I agree with them, that Arrow’s negative finding shows that an ordinal approach to welfare theory does not lead to satisfactory results (Edwards 1954 p. 390; Keeney & Raiffa 1976 }10.2.1). When von Neumann & Morgenstern (1944, 1947) presented their EU theory, with a utility unique up to level and unit (“cardinal”), the interest in the maximization of sums of individual utilities was revived. A famous result based on such a maximization is Harsanyi (1955), discussed in the preceding section. Such applications of cardinal utility for risk to welfare evaluation are, however, controversial. Many economists have argued that a cardinal utility index derived from risk behavior need not be the relevant cardinal index for welfare theory. Arguments to nevertheless identify these two indexes are in Wakker (1994). Blaug (1962) and Stigler (1950) presented the history of cardinal utility up to the middle of the preceding century. Abdellaoui, Barrios, & Wakker (2007 }3) updated this history. Young (1990) considered tax rates in various countries under the equal-sacrifice principle, entailing that every individual should sacrifice the same amount of utility to

92

Chapter 3: Applications of EU

tax. Under logarithmic utility this principle implies, for instance, proportional tax, where every individual should pay the same percentage of wealth. Young estimated the utility function that best explains tax rates imposed in various countries and, then, explicitly related this to utility functions estimated from decision under risk. He thus deliberately related utility derived under risk to (“riskless”) utility used in welfare evaluations.

3.9

Proofs for Chapter 3

Proof of Observation 3.3.1. First assume risk aversion. Then, in Figure 3.3.1, xj is preferred to the prospect qM(1q)m. By independence, the left prospect is preferred to the right. This implies aversion to elementary mean-preserving spreads. By transitivity, aversion to mean-preserving spreads follows. Conversely, assume aversion to mean-preserving spreads, and consider a prospect p1x1. . .pnxn with expectation m. The rest of this proof will demonstrate that the prospect is a mean-preserving spread of m, by reducing the prospect to m through a number of reversed mean-preserving spreads. Then, because of aversion to mean-preserving spreads, m ≽ p1x1. . .pnxn and we are done. We may assume that all pj’s are positive. Assume that there is an outcome that is different from m. Let j be such that pj|xjm| > 0 (the change in expected value when moving all probability mass from xj to m) is minimal; say, xj > m. There must exist an xi < m. Move the pj-probability mass from xj to m, reducing expectation. Also move pj(xjm)/(mxi) probability mass from xi to m, after which the expectation becomes m again. This can be done because pi  pj(xjm)/(mxi) by the definition of j. What has resulted is the prospect that assigns probability 0 to xj, probability pi  pj(xjm)/(mxi) to xi, extra probability pj þ pj(xjm)/(mxi) to outcome m, and other than that has the same probabilities as the original prospect. The original prospect is an elementary mean-preserving spread of the newly constructed prospect. The newly constructed prospect has one outcome more that is equal to m than the original prospect did. We continue the process. After at most n1 steps it stops because no outcome different from m can be left. Each prospect along the way was an elementary mean-preserving spread of the one to be constructed next, and was less preferred. The original prospect is a mean-preserving spread of m and m is preferred: risk aversion. □ Proof of Theorem 3.4.1. This result follows from Theorem 3.2.1a by expressing all payments in terms of utilities of person 1. In terms of these payments, person 1 is risk neutral and orders prospects by the expectation of these units. Person 2 being more risk averse than person 1 means precisely that, in terms of U1 units, person 2 prefers every prospect less than its expected value. That is, if payment is in U1 units, then person 2 is risk averse. The latter holds if and only if U2 is concave in U1 units. This is what should be proved. □

3.9 Proofs

93

Proof of Observation 3.5.1. This result was demonstrated by Keeney & Raiffa (1976 }4.10). That power utility (Eq. (3.5.3)) implies was also established in Exercise 3.5.1. □ Proof of Observation 3.5.2. This result was first presented and proved by Pfanzagl (1959). It can also be derived from Observation 3.5.1 through the substitution exp(a). Elaboration is omitted. The result is discussed by Keeney & Raiffa (1976 }4.8), but without a complete proof. Conversely, Observation 3.5.1 can be derived from Observation 3.5.2 through the substitution ln(a). Hence, Pfanzagl (1959) was the first to obtain these results. □ Proof of Theorem 3.7.3. See Fishburn (1965), or Keeney & Raiffa (1976 Theorem 6.4), or Wakker, Jansen, & Stiggelbout (2004 following Corollary 3.4). □ Proof of Theorem 3.7.5. See Keeney & Raiffa (1976 }5.4.3 and }6.3, “Remark Before Proof”). □ Proof of Theorem 3.7.6. See Anscombe & Aumann (1963). □

4

Expected utility with unknown probabilities and unknown utilities

In this chapter we return to general decision under uncertainty, with event-contingent prospects. As in Chapter 1, we assume that prospects map events to outcomes, as in E1x1


Enxn. Superscripts should again be distinguished from powers. EU is linear in probability and utility and, hence, if one of these is known – as was the case in the preceding chapters – then EU analyses are relatively easy. They can then exploit the linearity with respect to the addition of outcome utilities or with respect to the mixing of probabilities. Then the modeling of preferences amounts to solving linear (in) equalities. It was, accordingly, not very difficult to measure probabilities in Chapter 1 and to derive the behavioral foundation there, or to measure utility in Chapter 2 and to derive the behavioral foundation there. In this chapter, we assume that neither probabilities nor utilities are known. In such cases, results are more difficult to obtain because we can no longer use linear analysis, and we have to solve nonlinear (in)equalities. We now face different parameters and these parameters can interact. This will be the case in all models studied in the rest of this book. This chapter introduces a tradeoff tool for analyzing such cases. This tool was recommended by Pfanzagl (1968 Remark 9.4.5). Roughly speaking, it represents the influence that you have in a decision situation given a move of nature. 4.1

Fifteen preference questions

Answering the following questions by yourself will take about half an hour without you knowing at this stage what the questions will serve for. They are presented here, prior to explanations of the theories involved, because it is best to answer them according to your own subjective preference, without concerns about theories. We will use your answers later for measuring your utility and other concepts under various theories, and for preference foundations of those theories. Appendix K shows which parts of this book are based on this section (}}4.3, 4.4, 4.5, 4.6, 4.7, 4.8.2, 4.10, 4.11, 6.5, 9.4, 10.5, 12.3). The closer your answers are to your true preferences, the better the concepts calculated later will fit with you and the more interesting they are. There are no right or wrong answers to the questions, and they only and exclusively concern your subjective tastes. If you like to use calculations to 94

4.1 Fifteen preference questions

95

determine your answers then you can do so, but it is not necessary. Calculations are relevant only to the extent that you feel that they support your subjective preferences. An additional benefit of working on the following questions is that the structures of these questions will serve the development of concepts introduced later. If you plan to study the many analyses in this book of the following figures in detail, then you may make copies so as to have them easily available. If you prefer not to invest much time at this stage but still want to read sections based on this one, then you can proceed as follows: Skip the questions but, whenever data is needed, take the preferences of some subject described later – say, subject 5 in Table 4.3.1 – as default data. Fig. 4.1.1a illustrates the first question. $1000 is the intended unit of payment, but you may take in mind any other unit if you wish. Imagine that there will be elections in a foreign country with two candidates participating, of whom exactly one will win. The party of candidate 1, or cand1 as we will write more compactly (with cand2 similar), mostly won in the past, but this time around their campaign went badly. You have to decide on an investment in this country, the profits of which depend on which candidate will win. You have to make your decision now as is, without further information. When the outcome written on the dotted lines in Fig. 4.1.1a is, say, 11, then virtually everyone will prefer the right prospect in the figure, and when that outcome is 200 then virtually everyone will prefer the left prospect. That is, for low outcomes on the dotted line, the right prospect is preferred, and for high ones the left is preferred. In between the high and low outcomes there will be a switching value, where your preference changes from left to right, and where you have no clear preference. Call this value a1. If you have no clear preference for a whole region of outcomes, then the midpoint of that “region of indecisiveness” is taken as the best guess for the switching value a1. The measurement is chained, which means that an answer given to one question, such as a1 in Fig. a, is an input to the next question. The next question is in Fig. 4.1.1b, and the right prospect there has a1 as a possible outcome. You are again invited to determine the switching value on the dotted line, which is called a2. Obviously, a2 > a1. Similarly, Figs. c and d elicit a3 and a4. Now please fill out the figure. Figure 4.1.2 shows the following four preference questions. In Fig. 4.1.2a, the answer a1 given in Fig. 4.1.1a again serves as an input. Before asking the preference question, we first determine another input of the stimuli, namely the outcome g of the left prospect. For outcome g, choose a round number of approximately 1.5  a4. For the preference question, then determine the outcome G > g to be inserted on the dotted line such that the two prospects are indifferent. Because this experiment is chained, a misunderstanding in an early question can bias all subsequent data. For example, if you by accident took g ¼ 1.5  a1 as opposed to g ¼ 1.5  a4, then much will go wrong in measurements and analyses presented later in this book.

96

Chapter 4: EU; Probabilities and utilities unknown (a) cand1 wins

cand1 wins

...

10

∼ cand2 wins

1

cand2 wins

8

Your switching value on the dotted line is a1. (b) cand1 wins

cand1 wins

...

a1

∼ cand2 wins

1

cand2 wins

8

Your switching value on the dotted line is a2. (c) cand1 wins

cand1 wins

...

a2

∼ 8

1 cand2 wins

cand2 wins

Your switching value on the dotted line is a3. (d) cand1 wins

cand1 wins

...

a3

∼ 8

1 cand2 wins

cand2 wins

Your switching value on the dotted line is a4. Indicate in each figure which outcome on the dotted line ... makes the two prospects indifferent (the switching value).

Figure 4.1.1 [TO upwards] Eliciting a1 . . . a4 for unknown probabilities.

4.1 Fifteen preference questions (a) cand1 wins

cand1 wins

a1

10

∼ g cand2 wins

cand2 wins

⋅⋅⋅

Your switching value on the dotted line is G. (b) cand1 wins

cand1 wins

⋅⋅⋅

a1

∼ g

G

cand2 wins

cand2 wins

Your switching value on the dotted line is b2. (c) cand1 wins

cand1 wins

⋅⋅⋅

b2

∼ cand2 wins

g

cand2 wins

G

Your switching value on the dotted line is b3. (d) cand1 wins

cand1 wins

⋅⋅⋅

b3

∼ g cand2 wins

cand2 wins

Your switching value on the dotted line is b4.

Indicate in each figure which outcome on the dotted line ⋅⋅⋅ makes the two prospects indifferent (the switching value).

Figure 4.1.2 [2nd TO upwards] Eliciting b2, b3, b4.

G

97

98

Chapter 4: EU; Probabilities and utilities unknown (a) 0.5

⋅⋅⋅

α4

∼ 0.5

α0

Elicitation of γ 2. (b) 0.5

⋅⋅⋅

γ2

∼ 0.5

α0

Elicitation of γ1. (c) 0.5

⋅⋅⋅

α4

∼ 0.5

γ2

Elicitation of γ 3. Indicate in each figure which outcome on the dotted line ..., if received with certainty, is indifferent to the prospect.

Figure 4.1.3 [CEs] Eliciting g2,g1,g3.

In Figs. 4.1.2b4.1.2d, we always insert the values g and G just obtained. In Fig. 4.1.2b, determine the outcome b2 that makes the two prospects equivalent for you, and then determine b3 and b4 likewise. Figure 4.1.3 shows the next triple of preference questions. They concern probability-contingent prospects with given probabilities. The a’s are similar to those in Figure 4.1.1, with a0 ¼ 10. We elicit g2 before g1, in Fig. a. You are asked to give your certainty equivalents of the three prospects. The questions in Figure 4.1.4 are structured the same way as the preceding questions. Now the values elicited decrease. In Fig. a you determine a switching value d3 < a4, in Fig. b you determine a switching value d2 < d3; and so on. Figure 4.1.5 shows the final triple of preference questions. They again concern probability-contingent prospects with given probabilities. You are asked to indicate

4.1 Fifteen preference questions (a) cand1 wins

cand1 wins

...

a4

∼ cand2 wins

1

8

cand2 wins

Your switching value on the dotted line is d3. (b) cand1 wins

cand1 wins

...

d3

∼ 8 cand2 wins

cand2 wins

1

Your switching value on the dotted line is d2. (c) cand1 wins

cand1 wins

...

d2

∼ 8

1

cand2 wins

cand2 wins

Your switching value on the dotted line is d1. (d) cand1 wins

cand1 wins

...

d1

∼ 8 cand2 wins

1 cand2 wins

Your switching value on the dotted line is d0.

Indicate in each figure which outcome on the dotted line ... makes the two prospects indifferent (the switching value).

Figure 4.1.4 [TO downwards] Eliciting d3 . . . d0.

99

100

Chapter 4: EU; Probabilities and utilities unknown (a)

⋅⋅⋅ α1

α4

∼ 1–

⋅⋅⋅

α0

Elicitation of PE1.

(b)

⋅⋅⋅ α2

α4

∼ ⋅⋅⋅

1–

α0

Elicitation of PE2.

(c)

⋅⋅⋅ α3

α4

∼ 1–

⋅⋅⋅

α0

Elicitation of PE3.

Indicate in each figure which probability on the dotted line ... makes the prospect indifferent to receiving the sure amount to the left.

Figure 4.1.5 [PEs] Eliciting PE1, PE2, PE3.

which probabilities in the risky prospects generate indifference. Hence this method is often called the probability equivalent method. The abbreviation PE in the figure refers to this name. We will analyze your answers assuming EU in this chapter.

4.2

(Subjective) expected utility

This section gives elementary definitions for expected utility. Definition 4.2.1. Under Structural Assumption 1.2.1 (decision under uncertainty), expected utility (EU) holds if there exist a probability measure P and a strictly increasing utility (function) U : Ñ ! Ñ, such that

4.2 (Subjective) expected utility

E1 x1


En xn 7!

Xn j¼1

PðEj ÞUðxj Þ;

101

ð4:2:1Þ

the EU of the prospect, represents preference. □ In this model, the probability measure P and the utility function U are the subjective parameters that characterize the decision maker. The following exercise shows how to derive decisions from the EU model. Its elaboration is immediately given in the main text. Exercise 4.2.1.a Consider the vendor Example 1.1.1. Assume EU with P(s1) ¼ ¼, P(s2) ¼ ½, P(s3) ¼ ¼, U(400) ¼ 1, U(200) ¼ 0.7, U(100) ¼ 0.5, U(0) ¼ 0, and U(400) ¼ 2. Calculate the EUs of the four prospects in Table 1.1.1, and determine which prospect is chosen. □ In Exercise 4.2.1: EUðxÞ ¼ 1=4  1 þ 1=2  1=2 þ 1=4  ð2Þ ¼ 0; EUðyÞ ¼ 1=4  ð2Þ þ 1=2  1=2 þ 1=4  1 ¼ 0: EUð0Þ ¼ 0; EUðx þ yÞ ¼ 1=2  0:7 ¼ 0:35: Hence x þ y has the highest EU and is chosen. Exercise 4.2.2.a Consider the vendor Example 1.1.1 again. Assume EU with P(s1) ¼ ½, P(s2) ¼ 1/3 , P(s3) ¼ 1/6 , U(400) ¼ 1, U(200) ¼ 0.7, U(100) ¼ 0.5, U(0) ¼ 0, and U(400) ¼ 2. Calculate the EUs of the four prospects in Table 1.1.1, and determine which one is chosen. □ We next turn to the elicitation method for deriving theoretical constructs from observed choice. From now on this will be more complex than in the preceding chapters because all models in the rest of this book will involve at least two unknown functions, making it harder to identify each separately. In the model considered here, EU, the task is relatively easy because of additivity of probability. The elaboration of the following exercise is immediately given in the text. Exercise 4.2.3.a Consider a partition {E1,E2,E3}. Assume EU with the normalized U(0) ¼ 0 and U(100) ¼ 1, and assume 25E1 0  25E2 0  25E3 0 and 25  100E1 0 . What are P(E1), P(E2), P(E3), and U(25)? □ In Exercise 4.2.3, the first two indifferences imply PðE1 ÞUð25Þ ¼ PðE2 ÞUð25Þ ¼ PðE3 ÞUð25Þ; so that all three events have the same probability. These sum to 1 and must be 1/3 each. Then 25  100E1 0 implies U(25) ¼ P(E1)U(100) ¼ 1/3 . Hence, if we find equally likely events, then we can easily infer their probabilities and also measure utility. This was the basic idea of Davidson & Suppes (1956), Ramsey (1931), and Savage (1954).

102

Chapter 4: EU; Probabilities and utilities unknown

Exercise 4.2.4.a Consider a partition {E1,E2,E3}. Assume EU with the normalized U(0) ¼ 0 and U(100) ¼ 1, and assume 25E1 [E2 0  25E3 0; 25E1 0  25E2 0, and 15  100E1 0. What are P(E1), P(E2), P(E3), and U(15)? □ The probabilities in EU for uncertainty are sometimes called subjective. They need not be given beforehand and need not be based on objective grounds, although they may be (Exercise 2.3.1). They may describe subjective opinions of a person and may vary across different persons. In this respect they have the same status as utilities. Subjective probabilities appeared before in Chapter 1, where U was linear and expected value was maximized. In the literature, the term subjective expected utility, abbreviated SEU, is sometimes used instead of EU. As explained before, we will use the same terminology for risk and uncertainty whenever possible, so that we use the term EU for both contexts.1 The elicitation method worked out easily in the preceding two exercises, readily giving both probabilities and utilities. This was, however, due to luck because we had convenient equally probable events that made life easy. Such events are often not available in our observations, and there then is no very easy way to measure probabilities and utilities. Using the richness of outcomes as assumed throughout this book, the next section will present a way of measuring utilities and then probabilities that does work in general. Before turning to that, the rest of this section presents some preparatory results for EU. Exercise 4.2.5.!a Assume EU. Show that the following two claims hold. Together they amount to monotonicity as defined in Chapter 1. Assume that f ¼ E1x1. . .Enxn and g ¼ E1y1. . .Enyn. By Exercise 1.2.4 (partition flexibility), we may assume the same partition for both prospects. (a) Show that, if f(s)  g(s) for all s, then f ≽ g (weak monotonicity). (b) Show that, if f(s) > g(s) for all s, then f  g (strict monotonicity). □ We use the notation aE x

ð4:2:2Þ

to refer to the prospect yielding a under event E, and yielding the outcomes of x off E. With x ¼ (E1:10, E2:12, E3:8), for example, we have 15E1 x ¼ (E1:15, E2:12, E3:8) and 13E3 x ¼ (E1:10, E2:12, E3:13). A special case of this notation, aEb, when x is the constant prospect b, was used before. The notation aEx is most convenient if E is an outcome event of x, and we will usually restrict attention to such cases. For x ¼ E1x1· · ·Enxn and a general event E,

1

Several authors have used the term decision under risk whenever decisions are based on probabilities, be they objective as in Chapter 3 or subjective as considered here. Thus, they argued that Savage’s (1954) preference foundation of expected utility showed that decision under uncertainty can always be reduced to decision under risk. We will not follow this terminology but will use the term decision under risk exclusively for cases where the probabilities are objective and can be used as empirical primitives. Decisions based on nonobjective probabilities belong to decision under uncertainty and not to decision under risk in this book.

4.3 Measuring utility

EUðaE xÞ ¼ PðEÞUðaÞ þ

Xn j¼1

103

PðEj  EÞUðxj Þ;

where EjE denotes set-difference, which means that EjE contains the states contained in Ej but not those in E. An event E is null if aE x  bE x for all prospects x and outcomes a and b, and nonnull otherwise. Outcomes of null events can be ignored for determining the preference value of a prospect. Null events are considered to be impossible. To avoid triviality, we assume that there is a partition with two nonnull events, as in Structural Assumption 1.2.1 (decision under uncertainty). Exercise 4.2.6.!a Assume Structural Assumption 1.2.1 (decision under uncertainty) and assume EU. Show that an event E is null if and only if P(E) ¼ 0. □ The uniqueness of utility is the same for uncertainty as it was for risk, as partly illustrated by the following exercise. We will later see that probability is uniquely determined. Exercise 4.2.7.!a Show that, if EU holds with a utility function U, then it also holds with utility function U* ¼ t þ sU, where t 2 Ñ and s > 0. □ The rest of this section can be skipped by readers with little theoretical interest. ≽ is continuous if, for every partition (E1,. . .,En) of S, the preference relation restricted to prospects (x1,. . .,xn) with outcome events E1,. . .,En satisfies the usual Euclidean continuity condition: the sets {(x1,. . .,xn): (x1,. . .,xn) ≽ (y1,. . .,yn)} and {(x1,. . .,xn): (x1,. . .,xn) ≼ (y1,. . .,yn)} are closed subsets of Ñn for each (y1,. . .,yn). Wakker (1993a) demonstrated that this finite-dimensional continuity condition suffices for obtaining the traditional behavioral foundations. The implication of the following assignment can be reversed under nondegeneracy, so that U is continuous if and only if preference is (Wakker 1988). Assignment 4.2.8.b Assume EU. Show that ≽ is continuous if U is. □

4.3

Measuring utility and testing EU for }4.1

This section analyzes the choices in }4.1 in terms of EU. Relative to Chapter 2, an extra difficulty is that the probabilities need not be known. Cand1 abbreviates candidate1, and cand2 is similar. Exercise 4.3.1.!a Consider Figure 4.1.1 (TO upwards; do not consider the other figures). Assume that both candidates have a nonzero probability of winning. Show that, under EU (with a0 ¼ 10, p1 for the (subjective) probability of cand1 winning, and p2 ¼ 1  p1): Uða4 Þ  Uða3 Þ ¼ Uða3 Þ  Uða2 Þ ¼ Uða2 Þ  Uða1 Þ ¼ Uða1 Þ  Uða0 Þ: First derive the last equality using only Figs. 4.1.1a and b. □

ð4:3:1Þ

104

Chapter 4: EU; Probabilities and utilities unknown

Outcome under cand2

This point indicates the prospect (cand1:α1, cand2:8) 8

1

α0 (=10)

α1

α2

α3

α4 Outcome under cand1

Curves designate indifference.

Figure 4.3.1 Your indifferences in Figure 4.1.1.

The exercise has shown that a0,. . .,a4 are equally spaced in utility units. I am making this claim about your utility without knowing your (subjective) probability of cand1 winning. Under EU, your probability will obviously affect which outcomes a1,. . .,a4 you chose, but it apparently does not affect the correctness of the claim in the exercise. Whatever your a’s are, they are equally spaced in utility units. The pattern of Figure 4.3.1 clarifies why the term sawtooth method has sometimes been used for related measurement methods (Fishburn 1967 p. 450; Louviere, Hensher, & Swait 2000). If we normalize U(a0) ¼ 0 and U(a4) ¼ 1, which can always be done (Exercise 4.2.7), then the four equally big steps U(aj)  U(aj1), jointly bridging a distance 1, must each be ¼, and we can draw a graph of utility as in Figure 4.3.2. The figure shows that the measurement of the aj’s can be interpreted as the measurement of inverses of utility, with aj ¼ U1(j/4) for all j. 1 U ¾

½

¼

α0

α1

α2

α3

α4

=

0

10

Figure 4.3.2 Utility graph derived from Figure 4.1.1.

$

4.3 Measuring utility

105

Using refined versions of the measurement now explained, taking the “gauge” outcomes (8 and 1 in Figures 4.1.1 and 4.3.1) closer to each other, taking the starting value a0 lower if desired, and measuring more than four steps, we can measure utility to any desired degree of precision and extent. Next, with utility available, we can measure subjective probabilities using the same techniques as in Chapter 1, where in fact we assumed utility available (it being equal to money there). For example, for each pair of disjoint events we can measure the exchange rate of utility units between them, giving the proportion of their probabilities. And for each event we can measure how much utility for sure is equivalent to receiving one utility unit conditional on the event and zero utility units otherwise, after which the equivalent utility for sure is the probability of the event. Thus we can, in principle, completely measure subjective probabilities and utilities in full generality, be it that it takes more work than we needed in preceding chapters. We now turn to consistency checks of the measurements. Utility measurements similar to those inferred from Figure 4.1.1 can be derived for the questions in Figure 4.1.2 (2nd TO upwards). Figures 4.1.1 and 4.1.2 together provide a consistency test of EU, as part (b) of the following exercise demonstrates. Exercise 4.3.2.!a Assume EU for Figures 4.1.1 and 4.1.2, with nonzero probabilities of winning for both candidates. (a) Show that U(b4)  U(b3) ¼ U(b3)  U(b2) ¼ U(b2)  U(a1) ¼ U(a1)  U(a0). (b) Show that bj ¼ aj for all j. □ As the exercise shows, Figures 4.1.1 and 4.1.2 provide two different ways of measuring the same values aj ¼ bj under EU. If the equalities do not hold then either EU must be abandoned or the answers must be reconsidered, possibly using an error theory. Figure 4.1.3 (CEs) also provides a consistency test of EU when combined with Figure 4.1.1: Exercise 4.3.3.!a Assume EU for Figures 4.1.1 and 4.1.3, with nonzero probabilities of winning for both candidates. Show that gj ¼ aj for all j. □ If consistency checks reveal systematic inconsistencies, then for descriptive purposes nonexpected utility models can be developed to accommodate these inconsistencies. Figures 4.1.1 and 4.1.4 together test elementary properties of EU, as we will see next. These elementary properties also hold true for the nonexpected utility models introduced later, so that violations of EU found through Figures 4.1.1 and 4.1.4 cannot be accommodated by those models. Exercise 4.3.4.!a Do not assume EU (deviating from the title of this section). Assume only weak ordering of your preference relation. Further assume strong monotonicity, which means that any prospect becomes strictly better as soon as one of its outcomes is strictly improved. Under EU, not assumed here, the latter assumption

106

Chapter 4: EU; Probabilities and utilities unknown

would amount to all outcome events being nonnull. Show that dj ¼ aj for all j in Figures 4.1.1 and 4.1.4. □ Exercise 4.3.5.!a Assume EU for Figure 4.1.5. Throughout we normalize U(a0) ¼ 0 and U(a4) ¼ 1. Assume the data of Figure 4.1.1, and the implications of EU there. Do not consider your own answers PEj in Figure 4.1.5. Instead, consider the answers PEj that EU predicts given U(aj) ¼ j/4 for all j. Show that EU predicts PEj ¼ j/4 for all j. In other words, your answers in Figures 4.1.1 and 4.1.5 violate EU unless PEj ¼ j/4 for all j. □ Exercises 4.3.1–4.3.5 have demonstrated that, if EU holds deterministically, then all elicited values in Figures 4.1.1–4.1.4 should agree (aj ¼ bj ¼ gj ¼ dj for all j), and PEj ¼ j/4 for all j. You can verify whether your answers satisfy these equalities. A statistical analysis of these choice questions will be given later, based on the answers given by five students in individual interviews done in 1996. Table 4.3.1 contains their answers.

Table 4.3.1. Data of five subjects

a0 a1 a2 a3 a4 a0 a1 b2 b3 b4 a0 g1 g2 g3 a4 d0 d1 d2 d3 a4 PE1 PE2 PE3 g G

Subject 1

Subject 2

Subject 3

Subject 4

Subject 5

10 20 42 55 67 10 20 40 57 70 10 18 30 48 67 20 30 40 50 67 0.35 0.50 0.80 120 135

10 17 27 42.5 60 10 17 27 40 52 10 18 32 40 60 15 23 32 45 60 0.25 0.50 0.75 200 255

10 28 50 80 115 10 28 56 105 160 10 25 45 75 115 22 35 55 80 115 0.625 0.75 0.85 200 350

10 25 50 85 140 10 25 50 85 130 10 18 35 65 140 40 50 70 100 140 0.30 0.70 0.90 200 270

10 17 32 47 62.5 10 17 27.5 41 55 10 13.5 20 30 62.5 13 22 35 47 62.5 0.25 0.75 0.90 100 122

4.4 A decision process

107

The subjects in the experiment violate deterministic EU in many ways. Even, virtually all equalities implied by EU are violated. For example, a4 ¼ b4 is violated for every subject, and so is the equality aj ¼ gj for every subject and every j. Also d0 ¼ a0 is violated for every subject. These violations may of course result from randomness in the answers rather than from a genuine violation of EU. A statistical analysis will be given later. A first inspection of the data suggests that the differences between the b’s and the a’s are not systematic, that the g’s are mostly below the a’s, that the d’s mostly exceed the a’s, and that the PEj’s exceed j/4. If our sole purpose is to measure the utility function of a person, then it is usually easier to use events with known probabilities as in Figures 4.1.3 and 4.1.5 than events with unknown probabilities as in the other figures. The main reason for the use of unknown probabilities here is that this case best prepares for theories developed later. Assignment 4.3.6.a Assume EU with a positive probability of cand2 winning. Show that U(G)  U(g) ¼ U(8)  U(1). □ 4.4

A decision process

This section presents an interpretation of some indifferences in }4.1, and for the tradeoff tool used in most of this book. Parts of this section closely follow Ko¨bberling & Wakker (2004). The decision process for making the decisions in the experiment of }4.1 that best fits with the theories to be derived, and that matches the algebraic derivations of the exercises in }4.3, is as follows. We focus on the indifferences of Figs. 4.1.1a and 4.1.1d (TO upwards), writing E for the event of cand1 winning: a1 E 1  a0 E 8 and a4 E 1  a3 E 8: In both indifferences, the lower branches in Figs. 4.1.1a and 4.1.1d provide an argument in favor of the right prospect, because then 8 is received instead of 1, conditional on Ec (cand2 wins). For now, 81 is a linguistic abbreviation of “receiving outcome 8 instead of outcome 1.” It reflects the influence, i.e. the preference value created by the decision maker’s choice for the right prospect, if Ec is true. Thus, the mathematical notation reflects a component in the decision making process, in agreement with the homeomorphic approach of this book.2 In both indifferences, the upper branches provide an argument in favor of the left prospect, corresponding to a1a0 (receiving outcome a1 instead of outcome a0) in Fig. 4.1.1a and to a4a3 (receiving outcome a4 instead of outcome a3) in Fig. 4.1.1d, both contingent on E. Obviously, the strength of the argument in favor of the right prospect provided by 81 depends not only on how much better it is to receive 2

The notation  was suggested to me by Duncan Luce, as an analog of his joint receipt symbol

L .

108

Chapter 4: EU; Probabilities and utilities unknown

8 instead of 1, but also on event Ec, in particular its perceived likelihood. (The likelihood of) E is similarly relevant to the weight of the arguments provided by a1a0 and a4a3. At any rate, the indifference of Fig. 4.1.1a suggests the following interpretation: a1 a0 contingent on E exactly offsets 8  1 contingent on Ec : The indifference in Fig. 4.1.1d is interpreted similarly: a4  a3 contingent on E exactly offsets 81 contingent on Ec : These two interpretations together suggest that a4a3 can do the exact same thing as a1a0, i.e. offset 81 contingent on event Ec while itself being contingent on event E. We interpret this as a1a0 being an equally good improvement as a4a3, and use the notation a1 a0 t a4  a3

ð4:4:1Þ

as a linguistic abbreviation of this interpretation. The superscript t abbreviates tradeoff. The relation will be formalized in }4.5. 81 contingent on event Ec served as a gauge to establish Eq. (4.4.1). We will investigate the pros and cons of the interpretation given here in what follows, and see under which conditions it does or does not make sense. The interpretation given here appeals to a rational way of making decisions: by weighing arguments pro and con, while discouraging noncompensatory heuristics such as just going for certainty. 4.5

General utility measurement and the tradeoff notation

This section formalizes the concepts discussed so far for general prospects, and introduces a general tradeoff notation used in several parts of the book. The relationships t of Eq. (4.4.1) can be derived from other prospects than those just discussed. Figure 4.5.1 illustrates the following analysis. Consider general multiple-outcome prospects x, y, and consider the following two indifferences: aE x  bE y

ð4:5:1Þ

and gE x  dE y:

ð4:5:2Þ

Definition 4.5.1. We write a  b t g  d

ð4:5:3Þ

whenever we can find prospects x,y and a nonnull event E such that the indifferences in Eqs. (4.5.1) and (4.5.2) hold. The t relation is called t-indifference (abbreviating tradeoff indifference), or just indifference if it is clear from the context that tradeoffs are involved. □

4.5 Tradeoffs

109

Outcome under cand2

Prospect βEy y

E

c

xEc

δ

γ

β

α

Outcome under cand1 Curves designate indifference. α instead of β apparently offsets yEc instead of x c, and so E does γ instead of δ.

Figure 4.5.1 ab t gd.

The t-indifferences formalize the reasoning that was discussed in }4.4.3 In Eqs. (4.5.1) and (4.5.2), ab and gd, contingent on event E, offset the same gauge (receiving the x outcomes instead of the y outcomes) contingent on not-E. This suggests again that ab and gd are equally good improvements. The gauge now considered is more complex than gauges we saw in preceding sections. Now we allow that various outcomes and events are involved, but this does not alter the reasoning. In the notation of Eq. (4.5.3), we seem to be forgetting about E, x, and y. The e